Mixture Theory for Modeling Biological Tissues:
Illustrations from Articular Cartilage
Gerard A. Ateshian
Abstract Mixture theory has been used for modeling hydrated biological tissues for
several decades. This chapter reviews the basic foundation of mixture theory as applied to biphasic mixtures consisting of a porous-permeable deformable solid matrix
and an interstitial fluid. Canonical problems of permeation, confined compression
and unconfined compression are analyzed from theory and compared to prior experimental measurements on articular cartilage, with an emphasis on studies that
provide validations of theoretical predictions. A brief overview is also provided of
the application of mixture theory to solute transport, reactive kinetics, and growth
and remodeling.
Key words: Mixture theory; Biphasic theory; Articular Cartilage; Permeation;
Confined Compression; Unconfined Compression
1 Mixture Theory
Continuum modeling of biological tissues poses a number of challenges related to
the structure and composition of these tissues, and their temporal evolution as a result of biological and biochemical processes. Most biological tissues are anisotropic
and all soft tissues undergo large deformations. Similarly, most biological tissues
are porous and permeable, such that interstitial fluid pressurization and flow may
contribute significantly to their mechanics. In many cases, mass transport of soluble
species within the interstitial fluid plays an important role in the tissue’s metabolic
response. Electrically neutral and charged solutes, as well as charged molecular
species bound to the solid matrix of biological tissues, may also contribute to osmotic and electrical mechanisms, including pressures, potentials, flows, and curGerard A. Ateshian
Columbia University, New York, NY, USA, e-mail: ateshian@columbia.edu
1
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Gerard A. Ateshian
rents. Growth mechanisms, remodeling, and degradation all involve chemical reactions that alter composition, ultrastructure and properties of these tissues.
Mixture theory provides a continuum framework for modeling all these mechanisms and phenomena within a self-consistent formulation. For example, the solid
matrix of a biological tissue may be modeled as a heterogeneous mixture of solid
constituents, such as collagen, elastin and charged proteoglycans. Porous tissues
may be modeled as a mixture of a fluid and a solid, where the fluid itself may consist of a mixture of a solvent and multiple solutes. Chemical reactions among some
or all of these constituents may be incorporated to account for growth, remodeling
and degradation.
Mixture theory was initially formulated by Truesdell [83] and further extended by
a number of theoreticians in the 1960s and 70s [24, 26, 27, 28, 41]. The theoretical
application of mixture theory to biological tissues started in the mid-1970s [54] and
experimental investigations of biological tissues using this theoretical framework
began in earnest in the 1980s, most notably in the studies of articular cartilage by
Mow, Lai and co-workers [9, 56, 64, 68, 70, 71, 72].
One of the principal challenges in the adoption of mixture theory as a modeling framework for biological tissues has been the apparent complexity of its general formulation, especially when alternative traditional modeling frameworks already exist to describe various phenomena under specialized conditions. As noted
by Cowin [37], most papers that use mixture theory have an unusually large number
of equations. Indeed, mixture theory has a steep learning curve and a prospective
practitioner must balance the burden of its adoption against its potential benefits.
In this chapter, I hope to present the case for the benefits of using mixture theory,
by taking the reader through a narrative of the application and extension of this theory to the study of articular cartilage. This chapter neither provides an exhaustive
review of mixture theory, nor a complete review of the cartilage mechanics literature. My primary aim is to illustrate how mixture theory encompasses and combines
classical continuum mechanics frameworks and how it may extend those frameworks to accommodate challenges specific to biological tissues, and to demonstrate
experimental validations of its theoretical predictions.
2 Mass and Momentum Balance
Mixture theory is notoriously intimidating because it requires the formulation of axioms of mass, momentum and energy balance for each of the mixture constituents,
which may then be summed together to produce equivalent formulations for the mixture as a whole. Since mixture constituents may exchange mass, momentum and energy with each other, the constituent equations include interaction terms unfamiliar
to practitioners of classical continuum theories such as solid or fluid mechanics. In
a strict sense, the classical theories represent formulations for pure substances (e.g.,
a fluid consisting of only one substance). Truesdell conjectured that the mixture as
a whole should behave as a pure substance; this principle (which may be consid-
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
3
ered an axiom of mixture theory) places a constraint on the mass, momentum and
energy exchanges between constituents. Some of these constraints may be accepted
intuitively while others may seem unfamiliar in the context of classical continuum
mechanics. Importantly, as reviewed by Bedford and Drumheller [24], the axiom of
entropy inequality may only be applied to the mixture as a whole, or else it would
overly constrain the formulation of constitutive relations.
As with all continuum theories of heterogeneous substances, mixture theory assumes that all constituents coexist at every point in the continuum. In practice, this
mathematical assumption implies that all constituents coexist in a volume sufficiently small to encompass the relevant microstructure; each point in the continuum
thus represents the center of mass of that region.
Each constituent in an unconstrained mixture may move independently of other
constituents. The motion of constituent α in the mixture is given by χ α (Xα ,t),
where t is time and Xα is the position of a material point of constituent α in the
reference configuration of that constituent. In the current configuration at time t, an
elemental region whose center of mass is x = χ α (Xα ,t) contains material from all
constituents α, each of which may have originated from a different location Xα in
its reference configuration.
It is also possible to consider constrained mixtures where different constituents
move together in the current configuration. This type of mixture is most commonly
used to model solid constituents [51]. For example, in biological soft tissues such
as vascular wall or elastic cartilage, the extracellular matrix may consist of a constrained mixture of collagen and elastin. Constrained mixtures may also be used to
model discrete or continuous fiber distributions, where fiber bundles initially oriented in different directions are treated as distinct constituents α of a constrained
solid mixture.
The equations of mass and momentum balance are sufficient to address a broad
range of analyses in biological tissue mechanics. These general equations are presented below. The energy balance equations are not reviewed in this chapter, as
they are are only needed for more specialized analyses, such as those arising in bioheat transfer. The entropy inequality is needed to place constraints on constitutive
relations for functions of state, such as the stress, mass supply and dissipative momentum exchange for each mixture constituent. Since the formulation of these constraints is rather involved, only salient relations are summarized here, with proper
references to the prior literature provided for more interested readers.
2.1 Mass Balance
The axiom of mass balance for each constituent α of a mixture is given by
Dα ρ α
+ ρ α divvα = ρ̂ α
Dt
(1)
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Gerard A. Ateshian
where ρ α is the apparent density of constituent α, vα is the velocity of that constituent, and ρ̂ α is the apparent mass density supply to constituent α from all other
constituents. The apparent density ρ α owes its name to the fact that it represents
the mass of constituent α per mixture volume (both taken in the current configuration); similarly, ρ̂ α represents the mass supply to constituent α per mixture
volume. Sharing the mixture volume as a common denominator makes it possible to sum these parameters over multiple constituents. The operator Dα (·) /Dt =
∂ (·) /∂t + grad (·) · vα represents the material time derivative in the spatial frame,
following constituent α.
The mass balance for the mixture has the familiar form of the mass balance relation for a pure substance,
Dρ
+ ρdivv = 0
(2)
Dt
where ρ is the mixture density and v is the mixture velocity. The operator D (·) /Dt
is the familiar material time derivative in the spatial frame, following the mixture.
This equation is obtained by taking the summation of Eq.(1) over all α, defining the
relationships
ρ = ∑ ρα
(3)
α
for the apparent densities, and
v=
1
ρ α vα
ρ∑
α
(4)
for the velocities, and producing the constraint
∑ ρ̂ α = 0
(5)
α
in order to satisfy the requirement that the mixture as a whole behaves as a pure
substance. The variable ρ is the mixture density; it represents the mass of all constituents per mixture volume. The mixture velocity v represents the velocity of the
center of mass of the elemental region at x. Equation (5) simply states that any mass
gained by some constituent α must be due to mass lost from other constituents in
the mixture. It is an intuitively self-evident requirement in a Newtonian mechanics
framework.
2.2 Momentum Balance
The axiom of linear momentum balance for each constituent α is given by
ρ α aα = divTα + ρ α bα + p̂α ,
(6)
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
5
where aα = Dα vα /Dt is the acceleration of constituent α, bα represents external
body forces per mixture volume acting on constituent α, Tα is the apparent stress
in constituent α, and p̂α is the momentum supply to constituent α due to internal
momentum exchanges with all other constituents in the mixture. The apparent stress
owes its name to the fact that the associated traction vector tα on a plane with unit
normal n, tα = Tα ·n, represents the force vector acting on constituent α per mixture
area. Sharing the mixture area as a common denominator makes it possible to sum
traction vectors and stresses over multiple constituents. The momentum supply p̂α is
an internal body force that accounts for momentum exchanges among constituents.
These momentum exchanges may conserve or dissipate free energy, depending on
the presence of frictional interactions.
The momentum balance for the mixture has the familiar form
ρa = divT + ρb
(7)
where a = Dv/Dt is the mixture acceleration, and b is the mixture body force. This
form can be obtained by summing Eq.(6) over all constituents, so that the mixture
stress T is given by
T = ∑ Tα − ρ α uα ⊗ uα
(8)
α
where
= − v is called the diffusion velocity of constituent α. The resulting
constraint on the momentum supplies becomes
uα
vα
∑ p̂α + ρ̂ α uα = 0 .
(9)
α
In contrast to the mass balance equation, which produces intuitively self-evident
relations between the whole mixture and its individual constituents, the momentum
balance introduces less evident relations, such as that for the mixture stress T in
Eq.(8) or the constraint of Eq.(9). The unfamiliar term −ρ α uα ⊗ uα appearing in
Eq.(8) arises simply because ρa 6= ∑α ρ α aα in a heterogeneous mixture of unconstrained constituents, neither mathematically nor physically. In the special case of
a constrained mixture, where vα = v, ∀α, this term reduces to zero and the mixture stress equals the sum of constituent stresses, which is intuitively more evident;
however, the general case accounts for the fact that mixture constituents may have
a non-zero diffusion velocity that contributes a rate of change of linear momentum relative to the center of mass. Similarly, in the absence of mass exchanges
(ρ̂ α = 0, ∀α), Eq.(9) indicates that internal momentum exchanges should cancel
out, a familiar concept consistent with Newton’s third law of action and reaction.
However, in the presence of mass exchanges, such as those resulting from chemical
reactions between reactants and products, it is necessary to also account for the momentum loss from decreasing reactant mass and momentum gain from increasing
product mass.
The axiom of angular momentum balance reduces to Tα − (Tα )T = M̂α , where
M̂α is the skew-symmetric tensor whose dual vector represents the internal angular
momentum supply to constituent α due to interactions with all other mixture con-
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Gerard A. Ateshian
stituents. Assuming that the mixture as a whole models a non-polar material, the
constraint on this angular momentum exchange reduces to ∑α M̂α = 0. In applications of mixture theory to biological tissues, it is most common to also assume
that M̂α = 0, ∀α, as there is no compelling physical argument for assuming that
individual constituents behave as polar materials.
3 Biphasic Theory
A biphasic material is a binary mixture of a solid and a fluid constituent (α = s
and α = f ). Biphasic theory was formulated for the purpose of modeling biological
tissues as porous permeable deformable media. In this theory each constituent is
assumed to be intrinsically incompressible, there are no reactions between the solid
and fluid (ρ̂ α = 0 for α = s, f ), and isothermal conditions preclude heat flux. Biphasic theory is most appropriate for modeling biological tissues whose interstitial fluid
is mobile, such as cartilage [65, 69], intervertebral disc [45], bone [42], cornea [30],
or vascular tissue [46, 84]. The mobility of the interstitial fluid may be tested using
permeation experiments, which drive fluid through the tissue under the action of a
pressure gradient; or using osmotic loading experiments, which drive fluid into or
out of the tissue using osmolarity (chemical) gradients [21, 23, 30, 77].
Since water is nearly incompressible under physiological stress magnitudes, it is
reasonable to idealize the fluid constituent of a biphasic tissue as intrinsically incompressible. The assumption that the solid matrix may be idealized in this manner
must be verified experimentally, for example by measuring its volumetric change
under the action of a hydrostatic fluid pressure, as reported for articular cartilage for
pressures up to 12 MPa [22].
By definition, when a constituent is intrinsically incompressible, its true density
ρTα (mass of constituent α per volume of that constituent) is invariant in space and
time. The apparent and true densities are related by the volume fraction ϕ α of the
constituent (volume of constituent α per mixture volume) according to ρ α = ϕ α ρTα .
In a saturated mixture (a mixture with no voids), volume fractions satisfy the saturation condition
(10)
∑ ϕα = 1 .
α
For a biphasic mixture (α = s, f ), the relation for ρ α may be substituted into the
mass balance of Eq.(1) (with ρ̂ α = 0) and the resulting relations for the solid and
fluid may be summed, then simplified using Eq.(10) to produce
α α
div ∑ ϕ v
= 0.
(11)
α
This relation may be viewed as a reformulation of the mass balance for the mixture
in the special case when all constituents are intrinsically incompressible.
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
7
In mixtures that contain a solid constituent it is natural to define the boundaries of
the mixture on the solid. Therefore, the summation appearing inside the divergence
operator in Eq.(11) may be rewritten as ϕ s vs + ϕ f v f = vs + w, where
w = ϕ f v f − vs
(12)
is the volumetric flux of fluid relative to the solid (volume of fluid passing through
a cross-section of the mixture perpendicular to the flow, per time).
3.1 Constitutive Assumptions
The functions of state in a biphasic material are the stresses Tα , the internal momentum supplies p̂α , and the mixture free energy density Ψr (free energy of solid
and fluid in the current configuration, per volume of the mixture in the reference
configuration). By restricting our choice of state variables, we decide which material characteristics we would like to model. In biphasic theory we would like to
model the solid constituent as an elastic material, therefore we include the solid deformation gradient F as a state variable. We also would like to account for frictional
interactions resulting from the relative flow between fluid and solid, therefore we
also include the diffusion velocities uα in our list of state variables. However, we
are not interested in the frictional interactions within the fluid (viscosity) because
these can be shown to be negligible in comparison to frictional interactions between
constituents; therefore, we do not select the rate of deformation of the fluid as a state
variable. Similarly, we are not interested in modeling reactions between the fluid and
solid, therefore there is no need to include measures of solid and fluid mass content
(such as ρ α ) in the list of state variables.
This list of state variables is substituted into the axiom of entropy inequality by
expanding the material time derivative of the free energy using the chain rule of
differentiation. The assumption of intrinsic incompressibility of the constituents is
introduced using the method of Lagrange multipliers [57], by adding the product of
Eq.(11) with the multiplier p. The resulting expression for the entropy inequality
places the following constraints on the constitutive behavior of the mixture [13],
Ψr = Ψr (F) ,
(13)
1 ∂Ψr T
Ts = −ϕ s p −Ψ f I +
·F ,
J ∂F
T f = −ϕ f p +Ψ f I ,
(14)
(15)
p̂s = p gradϕ s + gradΨ f + p̂sd ,
(16)
f
+ p̂df ,
(17)
p̂ f = p gradϕ f − gradΨ
∑
α
p̂αd
α
·u ≤ 0,
(18)
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Gerard A. Ateshian
where Ψ f is the free energy density in the fluid (free energy per volume in the
current configuration) and p̂αd F, us , u f is the dissipative part of the internal momentum supply to constituent α. The expression of Eq.(18) is called the dissipation
inequality [35], as it represents the dissipation of free energy due to frictional interactions between the mixture constituents.
Equation (13) shows that the mixture free energy density only depends on the
solid deformation. Equations (14) and (15) provide the general relations between
constituent stresses and free energy densities, which require functional expressions
for both Ψr and Ψ f . Conveniently, the dependence on Ψ f goes away when we sum
the solid and fluid stresses and make use of the saturation condition in Eq.(10),
TI ≡ ∑ Tα = −pI +
α
1 ∂Ψr T
·F ,
J ∂F
(19)
where TI is called the inner part of the mixture stress. Thus, only a formulation for
Ψr is needed to evaluate the constitutive relation for the mixture stress. This expression also shows that the scalar multiplier p in the isotropic stress contribution, −pI,
represents the pressure in the interstitial fluid, since the remaining term only depends on the solid deformation. The convenience of using TI instead of Ts implies
that the mixture linear momentum balance in Eq.(7) is a more convenient alternative to the solid linear momentum balance in Eq.(6). Substituting the relations of
Eqs.(15) and (17) into the fluid linear momentum balance in Eq.(6) produces
ρ f a f = −ϕ f gradp + ρ f b f + p̂df ,
(20)
which conveniently does not include Ψ f either. Therefore, this reduced form of the
fluid linear momentum balance may be used, together with the mixture momentum
balance, to solve problems in the biphasic theory. (Alternatively, we may assume
constitutively that Ψ f = 0 on the basis that the free energy in the fluid is already
represented by the pressure p, as a proxy to free energy resulting from dilatation. In
that case, we recover the earlier biphasic theory formulation of Mow and co-workers
[70] and Holmes [47].)
Similarly, summing Eqs.(16) and (17) and making use of Eq.(9) in the absence
of mass exchanges, along with Eq.(10), produces
∑ p̂α = ∑ p̂αd = 0 .
α
(21)
α
This relation shows that the dissipative part of internal momentum supplies satisfy
the same constraint as the more general term. Combining the dissipation inequality
of Eq.(18) with the constraint of Eq.(21) shows that the general form for p̂αd is
[10, 57]
(22)
pαd = ∑ fαβ · uβ − uα ,
β 6=α
where fαβ (α, β = s, f ) are second-order tensors called frictional drag coefficients
[57, 70], which satisfy fβ α = fαβ .
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
9
For a biphasic mixture these
relations reduce to p̂df = −p̂sd = f f s · vs − v f , where
f f s is a function of F, us , u f in general; thus, the dissipative (frictional) momentum
exchange between fluid and solid is proportional to the relative velocity between
these constituents. Substituting this expression into the fluid momentum balance in
Eq.(20) produces the relation classically described as Darcy’s law,
h
i
w = −k · gradp + ρTf a f − b f ,
(23)
where
k= ϕf
2
ffs
−1
(24)
is called the hydraulic permeability tensor. The relation of Eq.(23) relates the relative fluid flux to its driving forces, namely, the gradient in fluid pressure and the
difference of inertia and body forces. In general, the permeability tensor k may depend on the state variables F, us , u f ; in a strict sense, Darcy’s law is recovered
when k is constant. Darcy’s law was originally formulated as a phenomenological
relation in porous media; mixture theory shows that it derives from the momentum
balance for the fluid constituent.
In practice, problems in biphasic theory may be solved by adopting several additional simplifications applicable to biological tissues. First, inertial effects are
typically neglected relative to other terms in the linear momentum balance, since
they are relevant mostly in wave propagation problems, where the assumption of
intrinsic incompressibility of the constituents would not be valid; thus, acceleration
terms involving aα are dropped out of those equations. Second, the diffusive terms
−ρ α uα ⊗ uα in Eq.(8) for the mixture stress are typically neglected in comparison
to the stresses Tα , as may be verified from an order of magnitude analysis using
typical stress, diffusive velocity and apparent density magnitudes expected to arise
in biological tissues; thus, the mixture stress and its inner part are assumed to be
the same, T ≈ TI . Finally, external body forces bα (typically representing gravity)
are only relevant in specific applications. Consequently, the most common usage of
biphasic theory employs the simplified expressions
divT = 0 ,
(25)
div (vs + w) = 0 ,
(26)
for the mixture momentum balance,
for the mixture mass balance, and
w = −k · gradp ,
(27)
for the fluid momentum balance (Darcy’s law).
The expression for T is approximated by TI in Eq.(19), which may be rewritten
as
T = −pI + Te ,
(28)
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Gerard A. Ateshian
where
1 ∂Ψr T
·F
(29)
J ∂F
is the stress resulting from solid matrix strain. This equation requires the formulation
of a constitutive relation for Ψr (F). A constitutive relation is also needed to describe
the dependence of k on F (its dependence on us and u f is neglected in practice). The
solid deformation gradient is uniquely related to the solid displacement u via F =
I + Gradu, where Gradu = ∂ u/∂ Xs and I is the identity tensor. The solid velocity is
also uniquely related to the displacement via vs = Ds u/Dt. Therefore, the unknowns
in a biphasic analysis are u and p, which may be solved from Eqs.(25) and (26),
using the relation of Eq.(27).
Many biological tissues undergo large deformations under normal physiological
conditions. Similarly, the solid matrix of many biological tissues exhibits anisotropy,
such that the constitutive relations for Ψr (F) and k (F) need to account for physiologically relevant material symmetries. Therefore, it is often necessary to solve
these biphasic equations using numerical schemes, such as the finite element method
[63, 82], that facilitate the solution of the resulting nonlinear equations.
Nevertheless, much insight may be gained into the response of biphasic materials
by obtaining analytical solutions under infinitesimal strains and rotations, assuming
that the solid matrix is isotropic. Under these conditions,
the deformation gradient
simplifies to F ≈ I+ε +ω, where ε = gradu + gradT u /2 is the infinitesimal strain
tensor and ω = gradu − gradT u /2 is related to the infinitesimal rotation tensor,
with gradu = ∂ u/∂ x representing the spatial gradient of the solid displacement. Under infinitesimal strains and the constraint of frame invariance, the relation of (29)
simplifies to Te = ∂Ψr /∂ ε. The solid matrix may thus be modeled using Hooke’s
law for isotropic elastic solids,
Te =
Ψr (ε) =
λs
(trε)2 + µs trε 2 ,
2
(30)
where the material constants λs and µs are the Lamé coefficients for the solid. It
follows from this relation that
T = −pI + λs (trε) I + 2µs ε
(31)
The simplest model for permeability assumes that it is isotropic and strain-independent,
k = kI
where k is a scalar material constant.
(32)
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
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3.2 Boundary Conditions
Boundary conditions are formulated by satisfying axioms of mass, momentum and
energy balance across interfaces in the mixture. Interfaces could represent external
boundaries of the mixture, or they may separate two regions of interest within the
mixture with a surface Γ . If this interface Γ is an idealization of a material surface (such as a thin membrane idealized as a mathematical surface), the interface
conditions need to allow for mass, momentum or energy jumps across that surface
(for example, surface tension in the membrane embodies a momentum jump). The
boundary conditions presented here apply to immaterial interfaces, such as boundaries of a biphasic tissue with its surrounding environment, or boundaries between
adjacent elements in a finite element mesh of a biphasic material.
3.2.1 Mass Balance
Since there are two sides across an interface Γ , we may denote them with + and −.
The outward unit normal to the + side is n+ and that of the − side is n− , such that
n− = −n+ . Assuming that each constituent is intrinsically incompressible and that
there are no reactions exchanging mass at the interface, the axiom of mass balance
for constituent α across Γ requires that
ϕ+α vα+ − vΓ · n+ + ϕ−α vα− − vΓ · n− = 0 ,
(33)
where the subscripts + and − represent quantities on either side of the interface
and vΓ is the velocity of the interface Γ [10, 41]. This expression summarizes the
requirement that the volumetric flux of constituent α normal to the interface must
be continuous across Γ . For convenience, let us define n ≡ n+ and [[ f ]] ≡ f+ − f−
for any argument f , so that Eq.(33) may be rewritten in a less cluttered form as
[[ϕ α (vα − vΓ )]] · n = 0 .
(34)
In a biphasic material, the interface Γ is typically defined to follow the motion
of the solid matrix, since the boundaries of a biphasic tissue are those of the solid.
In those cases we let vΓ · n = vs · n and we may examine boundary conditions for
three typical situations: At the interface between a biphasic material and a pure fluid
(ϕ s = 0 and ϕ f = 1), the jump condition of Eq.(33) as applied to α = f produces
ϕ f v f − vs · n = (v − vs ) · n, where v is the velocity of the pure fluid. This expression may be rearranged as
(vs + w) · n = v · n .
(35)
At the interface
between
two biphasic materials, the jump condition for the fluid
reduces to ϕ f v f − vs · n = 0, indicating that the volumetric fluid flux across
the boundary is continuous, which may be rewritten as
[[w]] · n = 0 .
(36)
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Gerard A. Ateshian
In this case, since vΓ · n is the same on both sides of Γ , it also follows that the
normal component of the solid velocity must be continuous across Γ ,
[[vs ]] · n = 0 .
(37)
Finally, at the interface of a biphasic material and a pure solid (ϕ s = 1 and ϕ f = 0),
Eq.(34) applied to α = f reduces to ϕ f v f − vs · n = 0, implying that the fluid on
the biphasic side may not flow across the interface Γ ; equivalently,
w·n = 0.
(38)
Note that there is no requirement imposed by the mass balance condition on
tangential components of the constituent velocities. Jump conditions on tangential
components may only be prescribed using constitutive assumptions. For example,
the tangential jump condition for the solid velocity between two adherent biphasic materials (a constitutive assumption) requires that (I − n ⊗ n) · [[vs ]] = 0. This
assumption, combined with the mass balance jump condition of Eq.(37), produces
[[vs ]] = 0 in the case of adhesive biphasic interfaces.
3.2.2 Momentum and Energy Balance
For an immaterial interface Γ , the jump condition on the momentum balance for the
mixture reduces to [10, 41]
[[T]] · n = 0 .
(39)
This condition is equivalent to requiring that the mixture traction vector, t = T · n,
be continuous across Γ . Letting T be given by the expression of Eq.(28), we may
also write t = −pn + te , where te = Te · n is the traction resulting from the solid
matrix strain.
The jump conditions for the momentum balance of the solid and fluid constituents involve the jump in internal momentum supply to these constituents. Similar to the expressions of Eqs.(16) and (17), these momentum jumps may not be
defined uniquely without further constitutive assumptions. Therefore, to complete
the set of boundary conditions, we must turn to the jump condition
derived
from the
energy balance for the fluid constituents [10], which reduces to µ̃ f = 0 where
µ̃ f is the mechano-electrochemical potential of the fluid (in units of energy per
mass). In the case of a biphasic material, the fluid is a pure substance (e.g., water)
which is electrically neutral, implying that its chemical and electrical potentials are
constants that may be set to zero with no loss of generality. In this case, µ f = p/ρTf
and the jump condition arising from the fluid energy balance reduces to
[[p]] = 0 ,
(40)
since ρTf is invariant for intrinsically incompressible constituents. This jump condition implies that the interstitial fluid pressure p of a biphasic material is continuous
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
13
across the interface Γ . If there is no fluid on either side of Γ , the jump condition of
Eq.(40) does not apply.
3.3 Permeation
Permeation is a canonical problem for biphasic materials, since it analyzes the
transport of interstitial fluid through the porous solid matrix and provides a direct
measure of the hydraulic permeability k. Permeation experiments are typically performed on a disk of tissue constrained within a tube with a rigid impermeable inner
wall (Fig. 1). The tissue specimen is placed against a free-draining rigid porous filter downstream of the flow. Optionally, the specimen is clamped upstream as well,
using another similar filter, to a pre-determined compressive strain. Either a known
fluid pressure is prescribed upstream (e.g., using a column of fluid), or a known
fluid velocity (e.g., using a syringe pump). Permeation experiments are notoriously
challenging because of the risk of leakage around the tissue specimen, which may
confound the true measurement of the tissue permeability. Another common challenge is that these types of experiments may take a long time to equilibrate to a
steady-state response; therefore, premature termination of the experiment may produce an unreliable measure of k.
v0
z=0
z
z=h
p0
upstream
fluid
biological
tissue
porous
free-draining
filter
downstream
fluid
Fig. 1: Permeation through a biological tissue.
An analytical solution to the permeation problem may help identify the conditions that alleviate some of these challenges, and may assist in interpreting the
results. For a permeation problem along the z-direction as shown in Fig. 1, a cylindrical coordinate system is adopted. For the one-dimensional axisymmetric conditions of this configuration, the only non-zero components of the displacement and
fluid flux vectors, u and w, are uz and wz , respectively, and the dependent variables
are only functions of z and t. Under these conditions, the mass balance in Eq.(26)
reduces to
14
Gerard A. Ateshian
∂
∂z
∂ uz
+ wz
∂t
= 0.
(41)
The fluid momentum balance in Eq.(27) simplifies to
wz = −k
∂p
,
∂z
(42)
and the mixture momentum in Eq.(25), combined with the constitutive relation in
Eq.(31), produces
∂ 2 uz
∂p
+ HA 2 = 0 ,
(43)
−
∂z
∂z
where HA = λs + 2µs is the aggregate modulus. Upstream, at z = 0, the boundary
conditions reduce to
∂ uz
∂ uz e
= va (t) , p (0,t) = pa (t) , Tzz (0,t) = HA
+ wz = 0 , (44)
∂t
∂ z z=0
z=0
where va (t) is the flow velocity upstream of the tissue sample and pa (t) is the
upstream pressure (Fig. 1). Downstream, at z = h, boundary conditions reduce to
∂ uz
uz (h,t) = 0 ,
+ wz = va (t) ,
∂t
z=h
(45)
∂ uz e
p (z,t) = 0 , Tzz (h,t) = HA
= σa (t) ,
∂ z z=h
where σa is the normal traction component between the tissue sample and the porous
filter. Here, we have made implicit use of the equation of continuity of mass for the
fluid entering and leaving the biphasic tissue, by requiring that the upstream and
downstream fluid velocities both be given by va . We also assume that because the
porous filter is free-draining, the downstream pressure is equal to zero, representing
atmospheric pressure. It should be appreciated that if va is known a priori, the upstream pressure pa and downstream traction σa can be determined a posteriori upon
completion of the analysis. Integrating the mass balance in Eq.(41) with respect to
z, and using the boundary condition either at z = 0 or at z = h, produces
wz = va (t) −
∂ uz
.
∂t
(46)
By eliminating ∂ p/∂ z from Eqs.(42)-(43) and using (46), we find that
HA k
∂ 2 uz ∂ uz
−
+ va (t) = 0 .
∂ z2
∂t
(47)
This is a partial differential equation in the unknown uz (z,t) alone. Once solved,
the fluid pressure can be obtained from the integration of Eq.(43) with respect to z,
making use of the boundary condition of Eq.(45),
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
∂ uz ∂ uz p (z,t) = HA
−
.
∂ z z
∂ z z=h
15
(48)
3.3.1 Steady-State Permeation
The steady-state response may be obtained from these equations by letting ∂ uz /∂t =
0 and va = v0 = constant. The mathematical steps leading to the solution are left to
the reader. The solution for the steady-state axial displacement is given by
z2
v0 h2
1− 2 ,
(49)
uz (z) =
2HA k
h
whereas that for the fluid pressure is
p (z) =
v0 h z
1−
.
k
h
(50)
Since the pressure varies linearly with z, it can be concluded that the pressure gradient is uniform through the thickness of the tissue sample at steady state. From this
expression it is now possible to determine the upstream fluid pressure at z = 0,
p0 =
v0 h
.
k
(51)
From an experimental perspective this is an important result because it shows that
the permeability can be determined from the measurement of p0 and the knowledge
of v0 and h, using k = v0 h/p0 . This result can also be substituted into the solution for
uz in Eq.(49) to express the solid matrix displacement as a function of the upstream
fluid pressure p0 ,
uz (z)
p0
z2
=
1− 2 .
(52)
h
2HA
h
The normal traction at the interface between the tissue and the porous filter is then
found to equal the upstream pressure in magnitude, σa = −p0 . The normal strain in
the axial z-direction is obtained from the slope of the displacement, and is found to
vary linearly through the depth,
εzz =
duz
v0
p0 z
=−
z=−
dz
HA k
HA h
(53)
The results of this steady-state permeation problem show that the pressure decreases linearly from p0 upstream to 0 downstream (Fig. 2a). The magnitude of
the upstream pressure is directly proportional to the fluid perfusion velocity and
sample thickness, and inversely proportional to the permeability. As the fluid flows
through the tissue, a drag-induced compaction occurs as indicated by the displacement profile (Fig. 2b). As a result, the height or thickness of the sample is reduced
16
Gerard A. Ateshian
p0
0
p0 h 2HA
0
− p0 HA
0
E zz ( z)
uz ( z)
p( z)
h
h
h
z
z
z
(a)
(b)
(c)
Fig. 2: Solution of steady-state permeation analysis, presented as a function of the
depth coordinate z.
by a magnitude of p0 h/2HA . This compaction is non-uniform, with the axial normal
strain starting at zero upstream, where the fluid pressure is highest, and increasing
linearly in magnitude with depth, achieving its highest value at the interface with
the porous filter, where the fluid pressure is smallest (Fig. 2c). The maximum strain,
which is compressive, is given by −p0 /HA ; clearly, for the above small-strain solution to remain valid, the upstream pressure p0 must remain small relative to the
tissue aggregate modulus HA .
When the axial normal strain (or, more strictly, the dilatation) changes in magnitude as shown in Fig. 2c, the assumption that the permeability remains constant
may not necessarily be valid experimentally and the above solution may need to be
reevaluated using a strain-dependent permeability function. However, it is of interest
that the above solution is in agreement with Darcy’s law, as long as the permeability
is assumed constant, considering that Darcy’s law does not address the deformation
of porous materials.
As a practical matter, in permeation experiments, the tissue sample needs to fit
tightly within the side wall of the test chamber to avoid compromising side-leakage.
This is sometimes achieved by osmotically swelling the sample after it has been
placed in the test chamber, but more frequently a clamping strain is applied onto the
sample via a second rigid porous filter placed upstream. (Oversizing the tissue sample relative to the diameter of the chamber and press-fitting it in place is generally
a less successful option, whereas the use of glue should be avoided due to seepage
into the tissue.) The analysis of a clamped sample would be similar to the above,
although the boundary conditions for the displacement function would be different.
As a final remark, the expression of Eq.(51) can be rewritten as
p0
v0
=
.
HA
HA k/h
(54)
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
17
from which it can be construed that just as HA may represent a characteristic measure of the stress in the tissue, so is HA k/h a characteristic measure of the interstitial
fluid velocity. For example, in articular cartilage, typical values of HA ∼ 0.5 MPa,
k ∼ 10−3 mm4 /N · s, and h ∼ 2 mm produce a characteristic velocity of 0.25 µm/s.
If the perfusion velocity is much smaller than this characteristic value, the upstream
pressure acting upon the tissue sample will be negligible compared to HA . Conversely, if v0 approaches this characteristic value, then p0 becomes non-negligible
relative to the aggregate modulus. Such analyses are helpful when designing an experimental apparatus and deciding upon the full scale range of the instrumentation,
such as pressure transducers or syringe pumps. Since v0 is the convective velocity
of the interstitial fluid whereas HA k/h is its characteristic diffusive velocity through
the mixture, the ratio of these two quantities is the non-dimensional Peclet number
for interstitial fluid flow through the tissue,
Pew =
v0 h
HA k
(55)
Usually, the Peclet number is invoked for transport of solutes in a solution (as the
ratio of convective to diffusive velocities) or for heat transfer (as the ratio of forced
convection to heat conduction). Here, we see that it is also applicable to fluid transport in porous media. Since Pew = p0 /HA in this problem, and since we already
explained that p0 should remain small compared to HA in order to keep the compressive strains small, it follows that Pew should also remain small compared to unity. In
practice, Pew . 0.2 is acceptable.
3.3.2 Transient Permeation
Permeation experiments on many biological tissues typically require a long time to
achieve the steady-state response described in the previous section, because of the
very low permeability of these tissues. To estimate the length of time required to
achieve steady state, it is necessary to solve for the transient response of uz (z,t),
either in response to a step increase in the perfusion velocity, va (t) = v0 H (t), or
a step increase in upstream fluid pressure, pa (t) = p0 H (t). The mathematical details for deriving the transient solutions for these two cases are not provided here,
though they are readily solved by standard methods for linear second-order partial
differential equations with constant coefficients.
When the velocity is prescribed upstream, the solution for uz (z,t) is
" #
uz (z,t)
z2
2 ∞ (−1)n
1
z −(n− 1 )2 π 2 t
w 1
τ
2
= Pe
1− 2 + 3 ∑
π e
,
cos n −
h
2
h
π n=1 n − 1 3
2
h
2
(56)
where the Peclet number Pew is given in Eq.(55) and
τ = h2 /HA k
(57)
18
Gerard A. Ateshian
may be called the gel time constant. The axial normal strain is then given by
εzz = ∂ uz /∂ z, and the fluid pressure may be obtained from Eq.(48). In particular,
the upstream pressure at z = 0 is given by
#
"
∞
2 2t
1
1
pa (t)
p (0,t)
2
−(n− 2 ) π τ
=
= Pew 1 − 2 ∑
.
(58)
e
HA
HA
π n=1 n − 1 2
2
Since the exponential term in these expressions will decay to zero as time increases
to infinity, it is easy to see that these equations reduce to the steady-state solutions
presented in the previous section. The complete transient response of the upstream
fluid pressure pa (t), normalized by its steady-state value, is plotted in Fig. 3a as a
function of normalized time. The pressure is found to increase monotonically with
time.
Let us address the question that first motivated this analysis: How long will it take
for the response to reach a steady state after initiation of the experiment? The easiest
way to address this question is to analyze the solution for the upstream pressure in
Eq.(58), since this pressure is typically measured in a permeation experiment where
va (t) is prescribed. Initially, at t = 0, this pressure is equal to zero. The steady-state
solution for the upstream pressure is pa (t → ∞) /HA = Pew ; in theory, according to
the solution, it will take an infinite amount of time to reach this steady-state value.
In practice however, we would be satisfied to stop the experiment after this upstream
pressure has reached perhaps 95% of its steady-state value. The characteristic time
constant for the increase in pressure can be deduced by looking at the first two terms
of the infinite series in Eq.(58). The time constants for these exponential functions
are given by
4
4
(59)
τ1 = 2 τ , τ2 = 2 τ .
π
9π
Since τ2 is nine times smaller than τ1 , the response is clearly dominated by τ1 . A
simple numerical calculation shows that the solution has reached 95 % of its steadystate value when
π2 t
8
(60)
0.95 ≈ 1 − 2 e− 4 τ , or t0.95 ≈ τ .
π
This calculation shows that the gel time constant τ provides a good estimate
of the time required to nearly reach steady state. In Section 3.3.1 typical values
of HA , k and h were suggested for articular cartilage. Using these values, we find
t0.95 ≈ 8000 s ≈ 2 h 13 m, which confirms that permeation experiments can be timeconsuming. Since the time constant is proportional to h2 , this time may be reduced
by a factor of four if the specimen thickness is halved.
When the fluid pressure is prescribed upstream, the solution for uz (z,t) is
!
uz (z,t)
p0
z2
4 ∞ 1 h
nπz i −n2 π 2 HA kt/h2
n
=
1 − 2 − 2 ∑ 2 1 − (−1) cos
e
(61)
h
2HA
h
π n=1 n
h
and the resulting fluid velocity across the tissue sample is
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
11
19
10
10
0.8
0.8
88
0.6
ppaa((tt)) 0.6
0.4
vv00hh kk 0.4
hv
hvaa((tt)) 66
0.2
0.2
22
pp00kk
00
44
00
00
0.5
0.5
11
t/τ
t/τ
1.5
1.5
22
00
(a)
0.5
0.5
t/τ
t/τ
11
(b)
Fig. 3: (a) Transient response of the upstream pressure pa (t) in a permeation experiment under a prescribed fluid velocity va (t) = v0 H (t). (b) Transient response of
the fluid velocity va (t) across the tissue sample in a permeation experiment with a
prescribed upstream fluid pressure pa (t) = p0 H (t). The gel time constant τ is given
in Eq.(57).
∞
2 2
2
h
va (t) = 1 + 2 ∑ e−n π HA kt/h
p0 k
n=1
(62)
Interestingly, we find that the initial velocity at t = 0+ is infinite, but eventually reduces to p0 k/h (Fig. 3b). This infinite value (which occurs because inertial effects
are neglected) arises from the fact that the initial fluid pressure p (z, 0+ ) increases to
p0 instantaneously throughout the tissue thickness, 0 ≤ z < h, except at the downstream porous filter (z = h) where the pressure must be zero according to the downstream boundary condition. Therefore, for an infinitesimal amount of time, there
exists an infinite pressure gradient gradp at z = h that produces an infinite fluid flux
wz = −k gradp at a fixed boundary where ∂ uz /∂t = 0.
The dominant time constant in the exponential decay of va (t) corresponds to
n = 1 in Eq.(62), and is given by τ1 = τ/π 2 . This value is four times smaller than in
the permeation problem with a prescribed upstream velocity, Eq.(59). Therefore, it
is more expedient to perform experiments with a prescribed upstream pressure than
a prescribed upstream velocity, as is also evident from a comparison of transient
responses in Fig. 3a and Fig. 3b.
3.3.3 Experimental Validations of Permeation
Permeation experiments have been reported for a number of connective soft tissues
such as articular cartilage [65], intervertebral disc [45], and ligament [87], as well
as for vascular tissue [46, 84] and hydrogels such as alginate and agarose [2, 5].
Since these experiments aimed to characterize the hydraulic permeability k of these
tissues, they all focused on analyzing the steady-state response to a prescribed fluid
20
Gerard A. Ateshian
pressure or velocity. By varying the clamping strain across the tissue specimen, they
reported the strain-dependent characteristic of k, typically exhibiting an exponential
decrease with increasing compressive strain magnitude. Therefore, other than confirming that these tissues were permeable to their interstitial fluid, these studies did
not provide direct validations of the biphasic theory from permeation analyses.
Since limited experimental data have been published for transient permeation,
we provide an experimental data set obtained from the permeation study of Albro
et al. on agarose hydrogels [5]. A disk of Type VII agarose (9% w/v, 1.5 mm thick)
was clamped at 15% compressive strain and subjected to an upstream fluid pressure
of p0 = 7.4 kPa using a fluid column. The volumetric flow rate of fluid transporting
across the gel was determined using time-lapse photography of the fluid meniscus
formed in a capillary tube connected to the downstream side of the flow chamber.
Experimental results reported in Fig. 4 show that the fluid flux decreased with time,
consistent with the theoretical prediction reported in Fig. 3b. The permeability extracted from the steady-state fluid flux using Eq.(51) was k = 3.4 × 10−3 mm4 /N · s.
The aggregate modulus was then obtained from a single-parameter fit of the transient response, producing HA = 35 kPa. For comparison purposes, direct measurements of HA for the same type and concentration of agarose exhibited a strong dependence on compressive strain, decreasing exponentially from 203 ± 8 kPa in the
limit of 0% compression, down to 58 ± 0 kPa at 15% compression. Therefore, under a clamping strain of 15%, the best-fit value for HA obtained from the permeation analysis was reasonably consistent with direct measurements, especially when
we recall that permeation produces a non-uniform compressive strain distribution
within the tissue sample as shown in Eq.(53) and Fig. 2. Thus, assuming linear superposition of the clamping strain and permeation response, the actual compressive
strain within the sample at steady state ranged from 15% upstream to approximately
28% downstream.
0.18
va(t)
(µm/s)
0.12
experiment
curvefit
0.06
0.00
0
2000 4000
t (s)
6000
Fig. 4: Experimental results and theoretical curve-fit of the fluid flux, using va (t) in
Eq.(62), in a permeation experiment performed on an agarose disk with h = 1.5 mm,
under a prescribed fluid pressure p0 = 7.4 kPa. Unpublished raw data is from the
study reported by Albro et al. [5].
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
21
3.4 Confined Compression
Confined compression problems represent some of the simplest problems for which
closed-form solutions exist in the biphasic theory. The basic assumption of a confined compression problem is that the kinematics of the solid and fluid constituents
are entirely one-dimensional. Typically, a cylindrical tissue sample of radius r0 and
thickness h is placed within a chamber of equal diameter, whose side wall is rigid
and impermeable. The bottom of the chamber may be either rigid impermeable or
rigid porous and the specimen is loaded with a rigid free-draining porous indenter
of diameter equal to that of the chamber, save for a clearance to avoid interference.
The free-draining nature of the loading indenter and, optionally, the bottom of the
chamber, is necessary to allow fluid to exude from the tissue as it is being compressed (Fig. 5). If these pathways were not provided, the biphasic theory would
predict no deformation of the tissue sample since each of the constituents are assumed intrinsically incompressible and the confined nature of the loading would
prevent any change in tissue volume. For practical purposes, tissue confinement can
only be maintained under axial compression, since tensile loading would reduce the
diameter of the cylindrical sample as its lateral surface recedes from the side wall
of the chamber, thereby violating the assumption of one-dimensional kinematics.
Thus one-dimensional problems of this kind are always understood to be confined
compression.
z=0
z=0
tissue
z=h
tissue
z=h
(a)
(b)
Fig. 5: Confined compression testing configuration. The cylindrical tissue sample is
placed in a chamber with rigid impermeable side wall and loaded by a free-draining
rigid porous indenter. (a) The bottom of the chamber is rigid impermeable; (b) the
bottom of the chamber is free-draining rigid porous.
If a prescribed static load is applied onto the indenter, the sample will deform
under this steady load as the fluid exudes from the tissue. This time-dependent response is known as creep. Conversely, if the deformation of the tissue is prescribed
at the indenter, the reaction force exerted on the indenter by the tissue will rise as
long as the deformation is increased, then will relax when the deformation is main-
22
Gerard A. Ateshian
tained constant. This time-dependent response is called stress-relaxation. Creep and
stress-relaxation confined compression are easy to implement experimentally and
are often used to characterize the material properties of biological tissues which can
be modeled with the biphasic theory. Clearly, the loading or deformation prescribed
at the indenter can be of a much more general nature than creep or stress-relaxation;
another popular testing configuration is to prescribe a sinusoidal displacement or
load at the indenter and to analyze the response of the tissue under steady state. This
dynamic loading, which can yield the frequency response of the tissue, may also be
used to extract its material properties.
We assume that the cylindrical tissue sample is homogeneous. The governing
equations for a one-dimensional problem are analyzed in cylindrical coordinates,
as performed in the permeation analysis presented in Section 3.3. Therefore, the
governing equations are the same as those presented in Eqs.(41), (42) and (43). For
the configuration of Fig. 5a, the boundary conditions at z = h are
∂ uz = 0 , wz (z = h,t) = 0 ,
(63)
∂t z=h
indicating that the solid velocity and relative fluid flux in the axial direction are equal
to zero at the bottom of the chamber at all times. These boundary conditions imply
that va (t) = 0 in Eq.(47). Since the displacement of the tissue at the bottom of the
chamber is constrained, one of the boundary conditions for this partial differential
equation is
uz (z = h,t) = 0 .
(64)
Under the loading indenter, either the displacement or the applied traction may be
prescribed. For a displacement-control experiment,
uz (z = 0,t) = ua (t) ,
(65)
where ua (t) is the prescribed displacement. For a load-control experiment, Tzz (z = 0,t) =
σa (t), where Tzz = −p + HA ∂ uz /∂ z is the total axial normal stress and σa (t) is the
prescribed traction, related to the applied load W (t) (assumed positive in compression) through σa (t) = −W (t) /πr02 . Because the loading indenter is free-draining,
the pressure of the interstitial fluid at the top surface of the tissue sample is equal
to the pressure of the fluid in the bathing solution. This is taken to be zero gauge
pressure,
p (z = 0,t) = 0
(66)
and thus the load-control boundary condition reduces to
∂ uz HA
= Ta (t) .
∂ z z=0
(67)
Once the solution for uz (z,t) has been obtained, the interstitial pressure throughout
the tissue sample can be obtained from Eq.(48), which represents the difference
in elastic stress between the location where the pressure is sought and the surface
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
23
under the indenter. Typically, the initial condition on uz (z,t) is that the sample has
no deformation at the beginning of the experiment, uz (z,t = 0) = 0.
3.4.1 Creep
In the creep problem, a step load is applied onto the tissue and maintained constant
as the tissue undergoes creep deformation. For this load-control case, the applied
traction may be specified as
σa (t) = σ0 H (t) ,
(68)
where σ0 = −W0 /πr02 is the constant traction corresponding to the constant applied
load W0 , and H (t) is the Heaviside unit step function. For this set of equations, the
solution for uz (z,t) is given by
0.05
0.04
uz 0,t
0.03
h
0.02
0.01
0
( )
σ0
= −0.05
HA
0
1
2
3
t/τ
Fig. 6: Creep deformation at the surface of a biphasic tissue under confined compression.
uz (z,t)
σ0
=
h
HA
(
z
−1
h
2 ∞ (−1)n
+ 2 ∑
sin
π n=1 n − 1 2
2
).
2 2t
1
z
1
n−
π
− 1 e−(n− 2 ) π τ
2
h
(69)
The axial normal strain is then obtained from εzz = ∂ uz /∂ z and the interstitial fluid
pressure may be evaluated from Eq.(48) as p = HA [εzz (z,t) − εzz (0,t)]. In particular,
at z = h, the fluid pressure is given by
σ0 2 ∞ (−1)n −(n− 1 )2 π 2 t
p (h,t)
=
∑ n− 1 e 2 τ .
HA
HA π n=1
2
(70)
Finally, the relative fluid flux may be evaluated from Eq.(46) as wz = −∂ uz /∂t.
Recall from Section 3.3.1 that HA k/h = h/τ is a measure of the characteristic
velocity of diffusive fluid flow within the biphasic matrix. We also note from the
24
Gerard A. Ateshian
exponent of the exponential function in the solution for the displacement that τ
is also a characteristic measure of the temporal response for creep problems. The
typical creep deformation response of confined compression is shown in Fig. 6,
where it can be observed that equilibrium is nearly reached at approximately two
and half times the gel time constant.
z,t))
εεzzzz((z,t
(( ))
z,t H
HAA
pp z,t
-0.05 -0.04
-0.04 -0.03
-0.03 -0.02
-0.02 -0.01
-0.01 00
-0.05
10−3−3
10
00
00
0.01 0.02
0.02 0.03
0.03 0.04
0.04 0.05
0.05
0.01
00
10−3−3
10
−2
−2
10
10
0.2
0.2
10−2−2
10
0.2
0.2
−1
−1
10−1−1
10
10
10
tt ττ
0.4
0.4
11
2.5
2.5
0.6
0.6
σσ00
−0.05
== −0.05
HAA
H
zz
hh
11
0.6
0.6
2.5
2.5
0.8
0.8
11
(a)
tt ττ
0.4
0.4
zz
hh
0.8
0.8
σσ00
−0.05
== −0.05
HAA
H
11
(b)
Fig. 7: (a) Axial normal strain εzz (z,t) and (b) interstitial fluid pressure p (z,t) in
confined compression creep, as a function of the axial coordinate z/h, at various
times t/τ.
The axial normal strain distribution is shown as a function of z and various times
t in Fig. 7a. It is apparent from this result that there exists a boundary layer near
the surface at early times, where the normal strain rapidly varies from the value
of σ0 /HA immediately under the porous indenter, to zero outside of the boundary
layer. However, as time progresses, the strain becomes more uniform with depth
until it reaches the constant value of σ0 /HA throughout the tissue at equilibrium.
A boundary layer is also observed in the spatial distribution of the interstitial fluid
pressure at early times (Fig. 7b). At the porous indenter the pressure is equal to
zero but at early times this pressure rapidly rises to the value of σ0 outside of the
boundary layer. Over time however, the pressure begins to decrease throughout the
tissue until it reaches the uniform value of zero at equilibrium.
The fluid flux, which is proportional to the gradient in pressure, is greatest at
early times and at the interface with the free-draining porous indenter and reduces
to a uniform value of zero at equilibrium. It can be noted that, instantaneously upon
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
25
loading, the relative fluid flux is infinite at the interface with the porous filter. However, immediately after that instant the relative fluid flux assumes finite values. The
equilibrium response for the creep problem can also be determined by taking the
limit of the solutions above as t → ∞, which reduces all the exponential terms to
zero,
σ0 z
σ0
uz (z,t)
=
− 1 , lim εzz (z,t) =
,
lim
t→∞
t→∞
h
HA h
HA
(71)
p (z,t)
wz (z,t)
lim
= 0,
lim
= 0.
t→∞ HA
t→∞ HA k/h
We find that the deformation is linear through the depth at equilibrium, the axial
normal strain is uniform, and the interstitial fluid pressure and relative fluid flux
reduce to zero. At equilibrium, a linear isotropic biphasic material behaves like a
linear isotropic compressible elastic material.
One note of caution when interpreting these results is the need to distinguish
between the observed responses described above and the assumed responses of the
tissue under physiological loading conditions. Most generally, soft hydrated biological tissues do not get loaded in situ via a porous indenter, therefore the boundary
layers in the strain, interstitial fluid pressure, and relative fluid flux observed in confined compression are generally not physiologic. This is of particular importance
when live tissue explants are tested to monitor their biosynthetic response to confined compression. Any biosynthetic activity observed near the interface with the
rigid porous indenter should be viewed as being specific to this choice of testing
configuration and not necessarily representative of the biosynthetic response of the
tissue in vivo.
3.4.2 Stress-Relaxation
In the stress-relaxation problem the indenter displacement is prescribed to increase
linearly in time and then kept constant until the tissue’s load response reaches equilibrium,
v0t t < t0
ua (t) =
,
(72)
v0t0 t > t0
where v0 is the indenter velocity during the ramp loading. For these equations, the
solution for uz (z,t) is
t
 z

h −1 τ

 2 ∞ (−1)n
2 2t


+ π 3 ∑ n3 sin nπ hz − 1
1 − e−n π τ
t < t0

uz (z,t)
n=1
t0
= −Pew
,
z

h

h −1 τ

∞
n
t
 2
2 2t
2 2 0

 + 3 ∑ (−1)
sin nπ hz − 1 e−n π τ en π τ − 1 t > t0
π
n3
n=1
(73)
where Pew is given in Eq.(55) and τ in Eq.(57). The axial normal strain εzz = ∂ uz /∂ z
may be evaluated from this solution, along with the interstitial fluid pressure p and
26
Gerard A. Ateshian
fluid flux wz . In particular, at z = h, the fluid pressure is

∞
1
4
−(2n−1)2 π 2 τt

t < t0

2 ∑
2 1−e
π
p (h,t)
n=1 (2n−1) .
= Pew
∞
t−t
1

−(2n−1)2 π 2 τ 0
−(2n−1)2 π 2 τt
HA
 42 ∑
e
−
e
t
>
t
0
2
π
(74)
n=1 (2n−1)
To evaluate the stress-relaxation response, the axial normal stress can be evaluated
at the interface with the porous indenter, σa (t) = Tzz (0,t) = −p (0,t) + HA ∂ uz /∂ z,

∞
2 2t
t

+ π22 ∑ n12 1 − e−n π τ
t < t0

τ
σa (t)
n=1
= −Pew
.
(75)
∞
2 2 t−t0
2 2t

HA
 tτ0 + 22 ∑ 12 e−n π τ − e−n π τ t > t0
π
n
n=1
11
0.1
0.1
−σ
−σaa(t)/H
(t)/HAA
0.05
0.05
p(h,t)/H
p(h,t)/HAA
00
00
0.5
0.5
t/τ
t/τ
(a)
11
−p(h,t)/σ
−p(h,t)/σaa(t)
(t)
0.5
0.5
00
00
0.5
0.5
t/τ
t/τ
11
(b)
Fig. 8: Confined compression stress-relaxation responses showing (a) the normal
compressive traction −σa (t) and interstitial fluid pressure at z = h, and (b) the interstitial fluid load support −p (h,t) /σa (t), when t0 /τ = 0.25 and Pew = 0.2.
The solution of Eq.(75) is presented for a representative case in Fig. 8a, with
t0 /τ = 0.25 and Pew = 0.2, such that the equilibrium compressive strain has the
magnitude v0t0 /h = 0.05. During the ramp phase, the axial normal stress increases
nonlinearly with time. At the end of the ramp, the stress relaxes to an equilibrium
value. The interstitial fluid pressure at the bottom of the chamber (z = h) similarly
rises during the ramp phase, then relaxes down to zero. By taking the ratio of the
fluid pressure to the total normal stress σa (t), the interstitial fluid load support can
be evaluated as shown in Fig. 8b. Initially, at t = 0+ , the fluid load support is 100%
immediately upon application of tissue deformation. This occurs because the fluid
has not yet had time to escape and the mixture acts as an incompressible fluid (or
solid) with uniform pressure and zero deformation. As time progresses however,
fluid exudes from the tissue and the interstitial fluid pressure and fluid load support
start decreasing with time. The strain profile through the depth of the tissue follows
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
27
a similar history to the creep response, with tissue compaction (increased strain)
occurring initially near the interface with the porous indenter and slowly progressing
to a uniform strain distribution at equilibrium. The fluid flux is also initially confined
to a narrow boundary layer near the porous indenter. As in the case of the creep
response, the equilibrium stress-relaxation response can be obtained by taking the
limit of the above solutions as t → ∞. We find that the resulting expressions have
the same form as in the creep problem,
V t
uz (z,t) z
0 0
=
−1
,
t→∞
h
h
h
p (z,t)
lim
= 0,
t→∞ HA
lim
lim εzz (z,t) = −
t→∞
V0t0
,
h
wz (z,t)
= 0.
t→∞ HA k/h
(76)
lim
If we compare the exponents of the exponential responses in creep and stressrelaxation, the time constant of the dominant term (corresponding to n = 1) is four
times greater in the creep problem than in the stress-relaxation problem. This means
that equilibrium is reached more slowly in creep than in stress-relaxation, which is
also evident when comparing the responses of Fig. 6 and Fig. 8. When designing an
experiment for testing biological tissue samples in confined compression, the shorter
duration of the stress-relaxation test may be considered beneficial, particularly when
attempting to minimize tissue degradation over long periods of testing.
3.4.3 Dynamic Loading
A frequent alternative to creep and stress-relaxation testing is dynamic loading in
confined compression. This testing configuration typically consists of prescribing
a sinusoidal load or displacement on the indenter and measuring the resulting response. To get the complete time-dependent response for this kind of loading, we
let
σa (t) = σ0 + σ1 sin ωt ,
(77)
where σ0 is a tare stress, σ1 is the amplitude and ω is the angular frequency of the
dynamic stress. This problem is simply the superposition of the creep solution of
Section 3.4.1 with the solution to the sinusoidal loading problem,
σa (t) = σ1 sin ωt ,
(78)
so we only need to solve the latter problem to complete the solution. The applied
traction must remain compressive at all times to ensure that the porous indenter
does not lift-off from the tissue; this constraint can be easily satisfied by having
|σ1 | 6 |σ0 |. The remaining boundary conditions are the same as those of the creep
problem described in Section 3.4.1.
The solution for the transient response under this dynamic loading configuration
is [80]
28
Gerard A. Ateshian
i
h z
uz (z,t)
σ1 ∞ (−1)n−1
=2
−
1
sin
α
n
∑
h
HA n=1 ω 2 τ 2 + αn4
h
(79)
,
2
−αn2 t/τ
× αn sin ωt − ωτ cos ωt + ωτe
where αn = n − 12 π. Plots of the transient displacement and corresponding fluid
pressure from this solution are presented in Section 3.4.4 below, where the solution
is compared to experimental measurements.
As a special case, it is also possible to get the steady-state response of the tissue
to dynamic loading by assuming that the√general solution at steady state has the
form uz (z,t) = udz (z, ω) eiωt , where i = −1 is the pure imaginary number. The
boundary conditions of Eqs.(64)-(67) have a similar form, with σa (t) = σ1 eiωt . It
follows that the solution for udz (z, ω) is given by
z
√
−
1
sinh
iωτ
d
uz (z, ω)
σ1
h
√
.
(80)
=
√
h
HA
iωτ cosh iωτ
Each term in the above expressions has been grouped such as to be non-dimensional,
e.g., udz /h, σ1 /HA , or ωτ. The solutions for the fluid pressure and relative fluid flux
can be similarly obtained.
0.05
0.05
(( ))
uuzdzd 0,
0,ωω
hh
0.001
0.001
σσ11
==−0.05
−0.05
H
HAA
0.025
0.025
ππ
(( ))
uuzdzd 0,
0,ωω
hh
00
0.1
10
0.1
10
ωω τ/2π
τ/2π
(a)
3π/4
3π/4
1000
1000
0.001
0.001
0.1
10
0.1
10
ωω τ/2π
τ/2π
1000
1000
(b)
Fig. 9: (a) Amplitude and (b) phase angle for dynamic confined compression under
load control, with σ1 /HA = −0.05.
The expression of Eq.(80) is a complex number whose magnitude represents the
amplitude of the response and whose argument is the phase angle. For example, the
amplitude and phase of the displacement response at the interface with the porous
indenter is shown in Fig. 9 as a function of the loading frequency f = ω/2π. At
very low frequencies, f τ −1 , the displacement is effectively in phase with the
applied load and its amplitude (given by the engineering strain measure udz /h) is
equal to |σ1 | /HA . (The figure shows a phase angle of π since the displacement
uz is positive when the prescribed compressive traction σ1 is negative, given our
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
29
choice of coordinate direction in Fig. 5a.) The load and displacement are in phase
because at very low frequencies there is plenty of time for the fluid to flow through
the tissue matrix and there is negligible drag between the two constituents. The
tissue behaves elastically with a modulus of HA . At very high frequencies, f τ −1 , the displacement is effectively π/4 out of phase with the applied load and the
amplitude of the deformation becomes negligible. This is because there is very little
time for the fluid to flow through the matrix as the load alternates back and forth;
with negligible exchange of fluid with the external environment the tissue acts as an
incompressible medium which cannot undergo any deformation in a rigid confining
chamber; its dynamic modulus theoretically tends to infinity as the frequency is
increased. For intermediate frequencies, f ≈ τ −1 , the tissue response is markedly
viscoelastic, with non-negligible relative fluid flow and biphasic drag forces and a
dynamic modulus greater than HA .
The interstitial fluid pressure can also be evaluated with this approach, producing
z
√
√
iωτ
cosh
−
1
− cosh iωτ
d
σ1
p (z, ω)
h √
=
.
(81)
HA
HA
cosh iωτ
3.4.4 Experimental Validation of Confined Compression
The biphasic theory was introduced by Mow et al. in two papers that appeared in
1980 [70, 68]. These papers provided theoretical solutions for confined compression
creep and stress-relaxation. Experimental results on bovine and human articular cartilage were also reported for these testing configurations. In the creep experiments,
an initial jump was observed in the displacement response which did not agree with
theory (Fig. 6) and was attributed to the lack of full initial confinement of the specimen within the test chamber [8, 70]. In stress relaxation, a successful comparison of
theory with experimental results on human knee cartilage was reported by Holmes
et al. [49] (Fig. 10), who extended the theory to account for strain-dependent permeability as motivated by earlier experimental findings [56, 65, 69]. These results
established that biphasic theory could successfully fit the response of articular cartilage in confined compression stress relaxation, which was a necessary condition for
validating the theoretical framework.
Since articular cartilage is a soft tissue that may undergo large deformations
in situ, finite deformation frameworks were subsequently formulated for biphasic theory to account for large strains [48, 55]. Ateshian et al. [14] performed
stress-relaxation, creep and dynamic loading experiments in confined compression
on bovine articular cartilage to investigate the theoretical framework proposed by
Holmes and Mow [48]. In their studies, stress-relaxation experiments were first
performed to curve fit the material properties of the tested specimen as follows:
Samples were compressed using five consecutive ramp-and-hold displacement profiles that each compressed the sample by 10% of its initial thickness, producing
five stress-relaxation responses to a final compressive strain of 50% (Fig. 11). The
elastic properties of the solid matrix were fitted from the equilibrium responses of
complete contact be made between the tissue
filter surface, Fig. 7. To do this, visual conta
articular surface and filter was made and then
Fig. 8(a) Nonlinear regression curve fit using the asymptotic expression for the stress-rise equation (25) for 0.6 < f < 1.0. The resultant
compression offset was added to assure
material parameters were used to predict the stress-relaxation stage for
tedigitation. If this is not done, spurious load
1.0 < f. This procedure provides a consistency check for the asympgenerated
which can lead to experimental arti
totic results. The asymptotic result, equation (25), is expected to fail
After complete stress relaxation occurs fr
nearf = 0 + .
conditioning (which takes approximately 45 m
function, equation (14), is imp
30
Gerard A.displacement
Ateshian
paring the theoretical and experimental results
the displacement and stress from the 5 perce
This was done because of the uncertainty of w
interdigitation can be achieved. However, if
o
include the offset in the calculations, only th
0.75
parameter k0 would be affected. In particular,
that there was complete interdigitation when
was made then k0 would be increased b
exp(0.05M).
CO
We have performed numerous stress relaxatio
o
o
using this protocol. A typical stress history is s
£ 0.25
r3
for the case of a slow rate of compression, w
CO
10 5 s and t0 = 5000s. The data shown are
removed from the lateral facet of a human pate
autopsy [37] from a 65-yr-old with no kno
osteoarthritis or other joint diseases.
Fig. 8(b) The material parameters determined from the asymptotic
To compare the biphasic theory with the
expressions,
see
insert,
are
used
in
a
numerical
scheme
to
determine
Fig. 10: Comparison
of biphasic theory and experimental measurements inresults
confined
it is necessary to determine the material
the entire stress-history over the 8000 s duration of this test. Note t0 =
compression
stress-relaxation,
as reported by Holmes et al. [49] (reproduced
withHA. The aggregate modulus is the ea
M, and
5000
s and e = 10 percent.
permission). The permeability was modeled to depend on the solid matrix dilatation
s
I s = trε according
= k0 eMI , where kEngineering
Journaltoofk Biomechanical
AUGUST 1985,
0 is the hydraulic permeability in the limit
of zero strain and M is a material parameter governing the dependence on the strain.
Downloaded 18 Jun 2012 to 128.59.144.110. Redistribution subject to ASME license or copyright; see http://www.a
these five steps (Fig. 11b). Then, the hydraulic permeability material constants from
a strain-dependent model were fitted to the transient response (Fig. 11a).
To validate this model and material properties obtained from fitting the stressrelaxation responses, a creep test was also performed on the same specimens, followed by dynamic loading at a frequency of 0.005 Hz. The fitted parameters from
the stress-relaxation response were used to predict the specimen deformation under
creep and dynamic loading under the same loading conditions. In this series of studies, no initial jump was observed in the creep response upon the application of the
step load, because an initial tare load was prescribed on the specimen to ensure full
confinement [14]. Very good agreement was observed between experimental results
and theoretical prediction (Fig. 12), providing strong support toward the validation
of the theoretical framework. The ability to predict outcomes of an experiment that
did not inform the model represents a sufficient step in the validation of a theoretical
framework.
In addition to predicting the deformation and stresses in the solid matrix, the
biphasic theory can also predict responses for the interstitial fluid pressure and flux.
As seen in Section 3.3.3, which reviewed experimental validations of biphasic permeation, there is a dearth of experimental studies that report the transient response
for interstitial fluid flux within cartilage. However, starting with the work of Oloyede
and Broom in 1991 [74, 73], experimental measurements of the interstitial fluid
pressure within cartilage have been reported. In the studies by Soltz and Ateshian
[78, 80], interstitial fluid pressure was measured in bovine articular cartilage at the
interface of the tissue sample and bottom of the confining chamber (z = h in Fig. 5a),
in creep and stress relaxation [78] and dynamic loading [80]. The biphasic theory
was used to extract HA and k by curve-fitting the tissue deformation at z = 0 (us-
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
31
Fig. 11: Experimental stress responses and theoretical curve-fit of confined compression stress relaxation on bovine articular cartilage, as reported by Ateshian et
al. [14] (reproduced with permission). (a) Complete transient response for five consecutive ramp-and-hold compression profiles that each compressed the sample by
10% of its initial thickness. (b) Equilibrium stress-stretch response from the end of
each compression step. The elastic properties HA0 and β of the solid matrix were obtained from fitting the equilibrium response. The hydraulic permeability parameters
k0 and M were obtained from fitting the transient response.
ing Eq.(69) in creep and Eq.(80) for dynamic loading experiments) or the stress
response of Eq.(75) in stress relaxation. The fluid interstitial pressure was then predicted from the theory, using Eq.(70) for creep, Eq.(81) for dynamic loading, and
Eq.(74) for stress-relaxation, using these values of HA and k. Very good agreement
was obtained between the predicted and measured interstitial fluid pressure in these
studies (Fig. 13), validating the ability of the biphasic theory to predict the transient
response of the interstitial fluid pressure in confined compression.
32
Gerard A. Ateshian
Fig. 12: Experimental displacement response of bovine cartilage plug under confined compression creep and dynamic loading (solid curve), and prediction of the
response from biphasic theory [48] using material constants fitted to the stressrelaxation response (Fig. 11). (Reproduced from [14] with permission.)
3.5 Unconfined Compression
Unconfined compression is a testing configuration that subjects a cylindrical tissue
sample to compressive strains in the axial direction and tensile strains in the radial
and circumferential directions. Since many biological soft tissues have a fibrillar
solid matrix that resists tension with much greater stiffness than compression, we
may extend the constitutive model of Eq.(31) to include the contribution of fibrils
that may only sustain tensile loading,
m
Te = λs (trε) I + 2µs ε + ξ
∑H
(i)
εn n(i) ⊗ n(i) ,
(82)
i=1
where ξ is the tensile modulus of each fibril bundle, m is the number of fibril bun(i)
dles, n(i) is a unit vector along the direction of the i−th fibril bundle, εn is the nor(i)
mal strain component along that bundle, εn = n(i) ·ε ·n(i) , and H (·) is the Heaviside
unit step function, which limits the contribution of the i−th fibril bundle to loading
(i)
configurations that produce a positive normal strain, εn > 0.
In the following analysis of unconfined compression the tissue is assumed to be
homogeneous and the loading platens are assumed frictionless. For the constitutive
model of Eq.(82), the governing equations for this problem can be reduced from the
general equations using cylindrical coordinates (r, θ , z) under axisymmetric conditions (zero circumferential displacement and fluid flux, and no dependence of the
remaining displacement components and fluid pressure on θ ). For simplicity, we
assume that there are only three fibril bundles (m = 3), each oriented along one of
the coordinate directions, such that the fibril directions n(i) coincide with the basis
vectors of this cylindrical coordinate system.
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
30
30
15
15
30
uuzz(0,t)
(0,t)30
00
(µm)
(µm)15
15
-15
-15
uuz(0,t)
z(0,t)
00
(µm)-30
(µm)
-30
-15 00
-15
-30
-30
30
30
00
15
15
30
uuzz(0,t)
(0,t)30
00
(µm)
(µm)15
15
-15
-15
uuz(0,t)
z(0,t)
00
(µm)-30
(µm)
-30
-15 00
-15
-30
-30
00
curvefit
curvefit
experiment
experiment
curvefit
curvefit
experiment
experiment
5000
5000 10000
10000 15000
15000 20000
20000
tt(s)
(s)
20
20
prediction
prediction
40
p(h,t)
p(h,t)40
00
(kPa)
(kPa)20
20
-20
p(h,t)-20
p(h,t)
00
(kPa)-40
(kPa)
-40
-20 00
-20
10000
10000
tt(s)
(s)
-40
-40
5000(a)
1000015000
15000
20000
prediction
prediction
experiment
experiment
5000
10000
20000
(s)
tt(s)
prediction
prediction
experiment
experiment
20
20
tt(s)
(s)
40
40
20
20
(s)
tt(s)
40
40
prediction
prediction
40
40
40
40
prediction
(b)
10000
00 prediction
10000
(s)
tt(s)
20
20
prediction
prediction
40
p(h,t)
p(h,t)40
00
(kPa)
(kPa)20
20
-20
p(h,t)-20
p(h,t)
00
(kPa)-40
(kPa)
-40
-20 00
-20
20
20
tt(s)
(s)
-40
-40
00
20
20
(s)
tt(s)
(c)
33
experiment
experiment
experiment
experiment
20000
20000
experiment
experiment
20000
20000
experiment
experiment
40
40
40
40
(d)
Fig. 13: Experimental and theoretical responses of bovine articular cartilage under confined compression dynamic loading, using data from the study of Soltz and
Ateshian [80]. (a) Experimental deformation and curve-fit of the tissue deformation uz (0,t) to Eq.(79), under the action of a dynamic compressive stress with
σ1 = −33 kPa and frequency f = 10−4 Hz, superposed over a static tare stress of
σ0 = −130 kPa; the fitted values are HA = 0.54 MPa and k = 1.6 × 10−4 mm4 /N · s.
(b) Experimental response and prediction of the fluid pressure p (h,t) for the same
specimen, using the values of HA and k from the curvefit in (a). (c) Experimental
response and prediction of uz (0,t) for the same specimen at a loading frequency
of 0.1 Hz, using properties from (a). (d) Experimental response and prediction of
p (h,t) for the conditions described in (c).
z
h/2
r
h/2
r0
Fig. 14: Geometry of unconfined compression problem.
34
Gerard A. Ateshian
We now make the following simplifying assumptions which anticipate the final
solution, in order to reduce the number of equations. These assumptions must be
consistent with the boundary conditions for this problem, which are described in
greater detail below. Because of the frictionless platens we expect the radial displacement ur to be independent of the axial coordinate z since no bulging of the
specimen is expected under these conditions (∂ ur /∂ z = 0). Since the shear strain is
given by εrz = (∂ ur /∂ z + ∂ uz /∂ r) /2 and is directly proportional to the shear stress,
and since the shear traction is zero on the top and bottom surfaces as well as the
lateral boundary, this suggests that we should also assume ∂ uz /∂ r = 0 everywhere
within the tissue sample. The loading platens are impermeable so that the fluid flux
normal to the platens must be zero, wz = 0. This constraint implies that the pressure gradient along z is zero at the top and bottom surfaces, and we assume it is
zero throughout the sample. Combining all these assumptions we get ur = ur (r,t),
uθ = 0, uz = uz (z,t), p = p (r,t), wr = wr (r,t), wθ = 0 and wz = 0. Finally, we anticipate from the nature of this problem that the axial normal strain εzz is compressive,
whereas the radial and circumferential normal strains, εrr and εθ θ respectively, are
tensile. Thus, only fibril bundles in the latter two directions contribute to the stress
response according to Eq.(82). Substituting these relations into the component form
of Eqs.(26) and (27), and the radial and axial components of Eq.(25) respectively,
we now get
∂ ur
∂ ∂ uz
1 ∂
r
+ wr +
= 0,
(83)
r ∂r
∂t
∂t ∂ z
∂p
wr = −k
,
∂r
∂p
∂ 1 ∂
−
+ H+A
(rur ) = 0 ,
∂r
∂r r ∂r
∂ 2 uz
= 0,
∂ z2
(84)
(85)
(86)
where H+A = HA + ξ combines the stiffnesses of fibrils and ground matrix in this
fibril-reinforced model. Integrating the last of these relations with respect to z, we
get ∂ uz /∂ z = ε (t), where ε (t) is the axial normal strain εzz in the cylindrical specimen, which is found to be only a function of time in this problem. Substituting this
result into Eq.(83) and integrating the resulting equation with respect to r yields
∂ ur
r2
r
+ wr = −ε̇ (t) + v (t) ,
(87)
∂t
2
where v (t) is an integration function. Evaluating this equation at r = 0 shows that
v (t) = 0 in this problem. Using Eq.(84), the above relation now reduces to
r
∂ p 1 ∂ ur
=
+ ε̇ (t)
,
(88)
∂r
k ∂t
2
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
35
which can be substituted into Eq.(85) to yield a partial differential equation in the
dependent variable ur (r,t),
∂ 1 ∂
r
1 ∂ ur
1
ε̇ (t) .
(rur ) −
=
(89)
∂r r ∂r
H+A k ∂t
H+A k
2
The boundary conditions for this problem must be formulated at all the boundaries
of the cylindrical tissue sample: at r = 0 and r = r0 (where r0 is the specimen radius),
and at z = ±h/2. Because of axisymmetry, there is no radial displacement or fluid
flux at r = 0 and the axial displacement is symmetric relative to the z−axis. These
conditions lead to the relations
∂ p e
= 0.
(90)
ur (0,t) = 0 , Trz (0,t) = 0 ,
∂ r r=0
The condition Trze = 0 is satisfied automatically throughout the cylindrical specimen
based on the assumptions summarized above; the condition ∂ p/∂ r = 0 is satisfied
automatically at r = 0 according to Eq.(88) as long as ur (0,t) = 0. At the radial
edge of the sample the total traction is zero, both in the normal and shear directions,
and the fluid pressure must be ambient,
∂ ur ur (r0 ,t)
e
Trr (r0 ,t) = H+A
+ λs
+ ε (t) = 0 ,
∂ r r=r0
r0
(91)
Trze (r0 ,t) = 0 ,
p (r0 ,t) = 0 .
At the top and bottom surfaces (z = ±h/2), the shear traction Trze is equal to zero
because of the assumption of frictionless contact, and the normal fluid flux must be
zero because the loading platens are impermeable, thus ∂ p/∂ z = 0; these boundary conditions are satisfied automatically based on our prior assumptions. For load
control experiments the integrated normal traction component at the top and bottom surfaces must be equal to the applied load, whereas for displacement control
experiments the axial displacement is prescribed,
( ´
r
2π 0 0 r −p + Tzze dr = W (t) load control
h
at z = ± .
(92)
2
displacement control
uz = ± 12 ua (t)
The interstitial fluid pressure is obtained by integrating ∂ p/∂ r in Eq.(85) and making use of the boundary condition on p in Eq.(91),
∂ ur ur r0
+
p (r,t) = − H+A
.
(93)
∂r
r r
The total normal load at the platens is then given by Eq.(92), W (t) = W p (t) +
W e (t), where
36
Gerard A. Ateshian
ˆ
W p (t) = −2π
ˆ
e
W (t) = 2π
r0
rp (r,t) dr
0
r0
.
(94)
rTzze (r,t) dr
0
W p (t)
Here,
is the component of the total axial load contributed by the interstitial
fluid pressure and W e (t) is the component contributed by the effective stress. Using
the above results, these expressions reduce to
ur ,
(95)
W p (t) = −πr02 λs ε (t) + (H+A + λs ) r r=r0
ur W e (t) = πr02 HA ε (t) + 2λs ,
(96)
r r=r0
H+A − λs ur W (t) = πr02 (HA − λs ) ε (t) −
.
(97)
HA − λs r r=r0
Note that the boundary condition of Eq.(91) was used to eliminate ∂ ur /∂ r|r=r0 from
the right-hand-side.
3.5.1 Instantaneous and Equilibrium Responses
For an unconfined compression stress-relaxation problem where the axial strain is
prescribed as a step function, let ε (t) = ε0 H (t), where ε0 6 0. The instantaneous
response at t = 0+ may be obtained by recognizing that p (r, 0+ ) is uniform (thus
wr (r, 0+ ) = 0) over the range 0 ≤ r < r0 , and ur (r, 0+ ) is a linear function of r over
that range. In that case, it can be shown that the instantaneous response is given by
lim
t→0+
ur (r,t)
ε0 r
=−
.
r0
2r0
It follows from this solution that
ε0
W p 0+ = πr02 (H+A − λs )
2
ε0 ,
+
2
W 0 = πr0 (2HA + H+A − 3λs )
2
(98)
(99)
so that the instantaneous fluid load support is given by
W p (0+ )
H+A − λs
2µs + ξ
=
=
,
+
W (0 )
2HA + H+A − 3λs
6µs + ξ
(100)
and the instantaneous (dynamic) unconfined compression modulus is
+
EY0 =
W (0+ )
1
= 3µs + ξ ,
2
2
πr0 ε0
(101)
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
37
whereas the instantaneous effective Poisson’s ratio is given by
+
ν 0 = lim −
t→0+
1
εrr
1 ∂ ur
= ,
= lim −
εzz t→0+ ε0 ∂ r
2
(102)
consistent with this instantaneous isochoric deformation. These results show that the
instantaneous stiffness of the tissue is significantly influenced by the fibril modulus
ξ : In the absence of fibrils (ξ = 0), the fluid load support in Eq.(100) reduces to
W p (0+ ) /W (0+ ) = 1/3 and the effective unconfined compression modulus reduces
+
to EY0 = 3µs ; however, as ξ increases to values much greater than the ground matrix
shear modulus µs , the fluid load support approaches unity,
W p (0+ )
= 1,
ξ /µs →∞ W (0+ )
lim
(103)
+
and EY0 increases in proportion to ξ /2. Thus, a relatively stiff fibril matrix has the
effect of enhancing fluid pressurization and interstitial fluid load support under instantaneous loading; as a result of this fluid pressurization, the effective compressive
modulus is nearly proportional to the tensile stiffness of the fibrils. This counterintuitive result implies that a hydrated biological tissue that is typically loaded in
compression, such as articular cartilage, may resist compressive loads more effectively by having a fibril-reinforced solid matrix, even though fibrils may only sustain
tension.
Similarly, the equilibrium response as t → ∞ is obtained by setting time derivatives to zero, recognizing that p = 0 and ur is similarly a linear function of r, thus
ur (r,t)
λs
r
=−
ε0 .
t→∞
r0
H+A + λs r0
lim
(104)
The effective equilibrium Young’s modulus in unconfined compression is
W (t)
2λs2
=
H
−
,
A
t→∞ πr 2 ε0
H+A + λs
0
E−Y = lim
(105)
and the corresponding effective equilibrium Poisson’s ratio is
ν− = lim −
t→∞
εrr
1 ∂ ur
λs
= lim −
=
.
εzz t→∞ ε0 ∂ r
H+A + λs
(106)
Since the fibrils may only sustain tension, the results presented here are specific to
unconfined compression. In particular, it may be noted from these last equations
that E−Y → HA and ν− → 0 as ξ /λs → ∞; thus, in compression, Young’s modulus
behaves as the confined compression modulus HA as the fibrils become very stiff,
consistent with the finding that the effective Poisson’s ratio tends to zero, implying
little lateral expansion under compression.
38
Gerard A. Ateshian
3.5.2 Transient Response
The transient solution for ur (r,t) may be obtained by the method of Laplace transforms and is given by


r
∞
J
γ
n
1
r0
2
1
−
η
ur (r,t)
r
= ε0 
+∑
(107)
e−γn t/τ 
r0
2η − 1 r0 n=1
γn J0 (γn ) (2η − 1 − η 2 γn2 )
where η = H+A / (H+A − λs ) and the gel time constant for this problem is given by
τ = r02 /H+A k .
(108)
ηγn J0 (γn ) − J1 (γn ) = 0 ,
(109)
Here, γn ’s are the roots of
where J0 and J1 are Bessel functions of the first kind, of order 0 and 1 respectively
(γn > 0). The fluid pressure may then be obtained from Eq.(93) using


r
∞ γn J0 γn r − 0 J1 γn r
r0
r
r0
2
∂ ur
1
−
η
e−γn t/τ  ,
(110)
= ε0 
+∑
∂r
2η − 1 n=1
γn J0 (γn ) (2η − 1 − η 2 γn2 )
and the fluid load support, evaluated from Eqs.(95)-(97), reduces to
∞
W p (t)
W (t)
2
J1 (γn )
e−γn t/τ
2
2
n=1 γn J0 (γn ) (2η − 1 − η γn )
(2η − 1) ∑
=
ηζ − 1 +
∞
2
1−η
J1 (γn )
e−γn t/τ
+∑
2η − 1 n=1 γn J0 (γn ) (2η − 1 − η 2 γn2 )
(111)
where ζ = 1 − HA /H+A .
A typical response for the time-dependent radial displacement ur (r,t) is presented in Fig. 15a as a function of time, for the case where η = 9/8 (or equivalently, H+A = 9λs ). The corresponding spatial distribution of the displacement is
presented in Fig. 15b at selected time points. Immediately upon loading (represented
by the very short time response t/τ = 10−4 ), an instantaneous lateral expansion of
the cylindrical specimen occurs, which varies linearly from 0 to r0 as predicted by
Eq.(98), after which the radial displacement slowly recoils to its equilibrium value
with increasing time.
The interstitial pressure for the same case is presented in Fig. 16a. The instantaneous response of the pressure is a homogeneous distribution whose magnitude is
given by −W p (0+ ) /πr02 in Eq.(99) (which evaluates to 4ε0 /9H+A in this example),
except at the boundary r = r0 where the pressure reduces to zero. Over time the
pressure becomes inhomogeneous and decreases toward zero, though it is noteworthy that the pressure near the center of the cylindrical specimen temporarily rises
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
0.05
0.05
(( ))
t
(( ))
uurr r,t
r,t
rr00
rr00
00
00
5
0.5
0.5
11
t/τ
t/τ
1.5
1.5
t/τ
22
0
0.5
1
t/τ
1.5
6
)
(a)
0.5
2
0
0
1.5
11
2
0
0.4 0.6
r/r0
(a)
0.8
1
0.5
2
0.4 0.6
r/r0
0.8
1
t/τ
10-2
10-4
( )
p r,t
H+ A
0.5
2
0
0
0.2
t/τ 10
10
10
( )
Wp t
H+ A
1
t/τ
0.2
0.2 0.4
0.4 0.6
0.6 0.8
0.8
r/r
r/r00
()
W (t )
( )
0.5
00
( )
(( ))
p r,t
0
00
10-4
( )
(( ))
(( ))
0
0.5
0.5
22
0.05
0.05
(b)
t/τ
t/τ
10-2
0.06
0.06 10
10-1-1
-2
-2
10
10
0.6
0.6
10
10-4-4
10-1 u compression
Fig.
15: Radial displacement in unconfined
stress relaxation under anur r,t
r0 ,t
ur r,t
r
W
W prp tt step strain ε0 = −0.1, with H+A =pp9λ
applied
r,t
r,t
rs0 , using Eq.(107). (a) Time-dependent r0
0.5
0
radial
displacement
along r, at selected time points
W
W tt displacement at r = r0 . (b) Radial
H
H++AA
2
0.5
0.5
t/τ. 0
0
0
22 0.5
0
1
1.5
2
2
0 0.2
0
0.2
0.4
0.6
0.8
1
00
00
t/τ
r/r
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
00
0.5
11
1.5
22
0.5
1.5
t/τ
r/r
t/τ
r/r00
t/τ
0.06 10-1
0.06 10-1
10-2
-4
0.6
10
0.05
)
t/τ
t/τ 10
10-4-4
10
10-2-2
10
10-1-1
0.05
0.05
uurr rr00,t,t
39
0
0.5
1
t/τ
1.5
2
(b)
Fig. 16: (a) Spatial distribution of interstitial fluid pressure in unconfined compression stress relaxation, at selected time points t/τ; and (b) interstitial fluid load support as a function of time when HA = 3λs .
above its instantaneous response before dropping down. The corresponding fluid
load support is given in Fig. 16b as a function of time, when ζ = 2/3 (or equivalently, HA = 3λs ), showing an initial jump, then a slow decrease toward zero as equilibrium is reached. For the choices of η and ζ selected in this example, Eq.(100)
predicts that the initial peak fluid load support is 2/3.
3.5.3 Experimental Validation of Unconfined Compression
The analytical solution for unconfined compression presented in the previous sections was first formulated by Armstrong et al. in 1984 [9] for the case ξ = 0 (no fib-
0
0.2
0.4 0.6
r/r0
0.8
1
Time (s)
ess relaxation load intensity time history from cond compression tests on GP specimens. Note how
onses coincide.
matrix with H^ = C^. The elastic and permeability coefficients
thus obtained for both GP and CE are provided in Table 1.
Significant differences were found in both elastic moduli (£3
and E\, p < 0.01) and permeability coefficients (p < 0.05)
ughout the tests, all specimens were bathed in between GP and CE (multivariate ANOVA). Growth plate is
NaCl solution containing protease inhibitors half as stiff as chondroepiphysis and twice as permeable. Pois40
Gerard A. Ateshian
son' s ratio (i^ai) was not significantly different. For comparison,
methylsulfonyl fluoride).
the isotropic model, Eqs. (39) and (40) in Armstrong et al.
alsothese
usedauthors
to curve-fit
thereport
experimental
data measurements
for
rils). In their(1984),
originalwas
study,
did not
experimental
onetheir
particular
specimen.
Curve-fits
the isotropic
and Singerman
to corroborate
theoretical
predictions.
In for
fact,both
in 1986,
Brown and
history of the load intensity in confined and transversely isotropic models are given in Fig. 7, in which the
reported
poor agreement
their the
measurements
on£1epiphyseal
(growth
ession stress relaxation tests of a GP [29]
specimen
biphasic
parameters between
that provided
best fit were:
= 4.3
cartilage
and£3the
response
of Armstrong
5. The load intensities at equilibriumplate)
(means
MPa,
= unconfined
0.64 MPa, stress-relaxation
v^^ = 0, i/ji = 0.49,
and fc,
= 5.0 X et al. [9].
ations) are plotted in Fig. 6. A highly
10"""
for the
isotropic remained
model, and
E =
In signifithose years,
them''/N-s
root cause
for transversely
this poor agreement
uncertain.
Eventuas found between the tissue levels (GP
versus
MPa,
= 0, and
= 15.5 X 10"'^
mVN-s emerged
for the isotropic
ally,
starting1.08
in the
late!v1990s,
two^ competing
hypotheses
that could explain
ficant difference was found betweenthis
thediscrepancy.
two model. The results indicate that the transversely isotropic model
d versus unconfined). Furthermore, consider- provides a much better fit to the experimental data than an
In suf1998, isotropic
Cohen, Lai
and Mow
[34] proposed
that the
unconfined
compression replate and chondroepiphysis specimens,
model.
In addition,
the optimized
solution
also yields
should
the of
known
disparity between
the tensile
and=compressive
power ( > 9 0 percent) exists for an sponse
inference
the account
requiredfor
value
the equilibrium
load intensity,
i.e.,/^,
m load intensities in the confined and
uncon- of the solid matrix of cartilage. Experimental studies had long demonproperties
tests are unlikely to differ by more strated
than one
that articular cartilage is much stiffer in tension than compression [1, 8, 53],
(paired Mest, a = 0.05, n = 20). Assuming
as
its
solid
matrix consists of fibrillar (type II) collagen that can resist tension, and
um load intensities are equal, an immediate Discussion
proteoglycans
(aggrecans)
that can model
resist compression
[66]. To account
The transversely
isotropic biphasic
provides an excelqs. (18) and (19) is that v^ = 0 for aggregating
both GP
forinthisthedisparity
in the context
of stress-relaxation
classical solid mechanics,
modeled the solid
lent description
of the
response inthey
the uncone following procedure was followed
fined compression
test,
where the tissue
is under
compression
in The stressmization procedure: First, v^j = 0 matrix
was preof articular
cartilage as
a transversely
isotropic
elastic
material.
the
axial
direction
and
tension
in
the
transverse
plane.
Previous
as given the value/e,/eo froni the unconfined
strain response in the axial direction employed the compressive modulus, whereas
Next, the elastic moduli Ei and i/2>, and the attempts to model this experiment with isotropic material propthat in the transverse
of isotropy
employed
modulus.
erties couldplane
not produce
adequate
results,the
as tensile
demonstrated
by They pron the transverse r-9 plane) were calculated
vided an analytical
solution
for unconfined
compression
the inherent
maximum
theoretical
limit of 1.5stress-relaxation,
on the ratio of similar to
meter curve-fit of the unconfined compression
the peak
equilibrium
intensity (Armstrong
et al.,
1984). epiphyseal
Eqs. (20). The axial permeability k^
(inofthe
that
Eq.(107),
thatto better
fitted load
the experimental
response
of bovine
Using
btained from results of the confined comprescartilage (Fig.
17).the transversely isotropic model for unconfined comprese easily shown that the confined compression
for the transversely isotropic solid matrix is
same equation as that for an isotropic solid
1.4
I
* experimental
•
•
transversely isotropic
—
0.20
isotropic
1.2
1.0
V9
0.8
4)
OR
0.4
L
a
0.10
r
*
/ '
1
L (
r'
'T*
^
^
/ f
L f
0.2
0.0
t^
r '
LI
0.00
200
400
600
time (s)
800
1000
Fig. 7 A typical stress-relaxation time history in response to a ramped
ensity at equilibrium for both growth plate (GP) and
Fig. 17: Experimental
response and theoretical curve-fits for unconfined compresdisplacement (10 percent compression at 131 s) with curve-fits of both
CE) in confined and unconfined compression tests
, n = 10)
isotropic and
Isotropic biphasic
models
sion stress-relaxation
oftransversely
bovine epiphyseal
cartilage,
as reported by Cohen et al. [34]
echanical Engineering
(reproduced with permission). Using a transversely isotropic model for the solid maAUGUST
1998,theVol.
120 / model.
495
trix of cartilage produced significantly better
fits than
isotropic
3 to 128.59.144.110. Redistribution subjectInto1999
ASME
licenseetoral.
copyright;
see concerns
http://www.asme.org/terms/Terms_Use.cfm
Bursac
[31] raised
about the use of a transversely isotropic
model to model articular cartilage, as this approach would produce inconsistent
results between confined and unconfined compression. In the same year, Soulhat
et al. [81] and Li et al. [61, 62] proposed to model the solid matrix of biphasic
unconfined compression "Fig. 5#. The shear modulus & was obtained from Eq. "31# using the equilibrium torque response T,
given the specimen radius r 0 and thickness h.
the material
parameters
been determined
from curveMixture Theory Once
for Modeling
Biological
Tissues:had
Illustrations
from Articular
Cartilage
41
fitting, the fluid pressure at the articular surface of the specimen
subsequently
predicted from
for unconfined
compres-material. In
cartilage in was
unconfined
compression
usingtheory
a fibril-network
reinforced
sion using Eq. "22# at r!0, and compared with the corresponding
their approach,
collagen data
fibrilstowere
using ability
linear or
experimental
test modeled
the predictive
of non-linear
the model.springs that
could only sustain
tension,
as showncomparisons
in Section 3.5.
Theyexperimental
demonstrated
Curve-fits
and predictive
between
andgood curvefits of the stress-relaxation
of articular
single
theoretical resultsresponse
were assessed
with a cartilage,
nonlinear under
coefficient
of or multiple
determination
r 2 '24,46(.
ramp-and-hold
displacement
profiles. Their formulation did not exhibit the limita-
tion raised in the study of Bursac et al. [31].
In 2000, Results
Soltz and Ateshian [79] adopted the Conewise Linear Elasticity (CLE)
framework formulated
earlierdev.
by Curnier
al. material
[38] forproperties
modeling(n!9)
elastic solids that
The mean$std.
values ofetthe
"16
were found
to be:
MPa and kto
exhibit different
behaviors
in Htension
and compression,
model the
solid matrix
"A !0.64$0.22
z !3.62%10
ntal confined compression stress-relaxation $0.97%10"16 m4/N•s from curve-fitting the transient confined
of
biphasic
cartilage.
They
performed
confined
and
unconfined
compression
and
2
r 0 ‡ and corresponding theoretical curve-fit compression response "r 2 !0.95$0.03, Fig. 4#, H !13.2
#Atheir compressive,
torsion experiments on bovine articular cartilage disks to extract
men
% 2 !0.48$0.23 MPa, and k r !6.06%10"16$2.10
tensile and $1.7
shearMPa,
properties,
as well as axial and radial permeability coefficients.
%10"16 m4/Ns from curve-fitting the transient unconfined comThey also measured
the
interstitial
fluid pressure at the center of the disk in un2
pression data "r !0.99$0.002, Fig. 5#. A paired two-tailed t-test
g platens. As for stress-relaxation confined
tests, the compression
stress
relaxation,
and compared these measurements to prepressed between the platens to a tare load of performed on the radial and axial permeability yielded p
dictions
from
this
biphasic-CLE
model.
These
results
good agreement
!0.001.
The
model
predicted
the
transient
fluidshowed
pressurevery
response
g tare equilibrium, a step rotation of ! 0
2
with
an
r
!0.98$0.01
in
unconfined
compression
"Fig.
5#.
The
between
theory
and
experiments
(Fig.
18),
providing
strong
support
for the validaplied about the axis of the cylindrical specimen
shear theory
tests produced
& !0.17$0.06
Tablefor
1 summarizes
the
tion Ann
of biphasic
in a framework
that MPa.
accounts
the tension-compression
tor "Slo-Syn Model 440, Warner Electric,
properties
obtained
for each
specimenusing
by curve-fitting
resultant equilibrium reaction torque,nonlinearity
T, was material
of articular
cartilage.
Further
validation
interstitialthe
fluid pressure
experimental responses, as well as r 2 values for the curve-fitting
torque transducer; for all specimens,
torque
measurements
in
unconfined
compression
was
also
reported
subsequently
in the
of the confined compression load response "c.c.f.# and unconfined
hieved within 60 seconds.
study of Park
et al. [75],
who
showed
that the
magnitude
fluid
compression
load
response
"u.c.f.#,
andpeak
for the
predictionofofinterstitial
the
For confined and unconfined compression,
a fluid
load support
in pressure
unconfined
compression
approached
100%
the conratio of tensile
in unconfined
compression
"u.c.p.#.
The as
elastic
-fitting algorithm was used to find the
stantsmoduli
provided
in Table
(H "A increased,
,H #A ,% 2 , &consistent
) representwith
a comto best-fit
compressive
of the
solid 1matrix
Eqs.(100) and
s by matching the experimentally measured plete set of properties for a material with cubic symmetry; these
(103).
force with the corresponding theoretical re-
r "23##. For computational efficiency, the theoere determined using a numerical finite differlving the governing differential equations "Eq.
ral difference was used for spatial differentiadifference in time, yielding a fully implicit
l solution domain "0$z$h for confined com$r 0 for unconfined compression# was divided
vals and the finite difference solution was obe step using a linear solver for band-diagonal
ons. Numerical integration was employed for
action force in unconfined compression, Eq.
ezoidal rule. Curve-fitting was performed using
ewton optimization method for minimizing a
ple bounds using a finite-difference gradient
merical Libraries, Visual Numerics, Houston,
ctive function given by the root-mean-square of
between the experimental and theoretical load
Fig.
5 Experimental
unconfined
compression
stress-
Fig. 18:
and theoretical responses for unconfined compression stressH "A and k z were obtained by curve-fitting
theExperimental
relaxation response † F u „ t …Õ ! r 02 ‡ and corresponding theoretical
relaxation
of
a
bovine
articular
cartilage disk, as reported by Soltz and Ateshian
e F c (t) to the confined compression solution curve-fit for the
same specimen as in Fig. 4. The experimental
withpressure
permission).
transient
axial
response
s value of H "A in Eq. "23#, H #A , % 2[79]
, and(reproduced
k r interstitial
at theThe
specimen
center
† p stress
„ r Ä0,t …‡
and cor-was fitted to
responding
theoretical
are also
presented.
urve-fitting the total load response F uextract
(t) frommaterial
parameters
for theprediction
biphasic-CLE
model
adopted in that study. The
DECEMBER 2000
interstitial fluid pressure was measured at the bottom center of the disk, showing
very good
agreement with the fluid pressure predicted from the model using the
Table 1
fitted material parameters.
Transactions of the ASME
ARTICLE IN PRESS
C.-Y. Huang et al. / Journal of Biomechanics 38 (2005) 799–809
42
Table 2
Gerardregions
A. Ateshian
Compressive properties of human glenohumeral cartilage for different
and zones
Zone measurements of
Region
A0
At the time of these studies, all tensile
cartilage propertiesHhad
been performed
using
specimens harvested
parallel to the Anterior
articulara surface, whereas
Humeral
head
Superficial
0.11070.030
a
Center
0.08370.038
compressive properties had been measured on disks harvested
with
their axis normal
a
Inferior
0.13970.064
to the articular surface. Thus, tensile and compressive moduli
reported
in
the
prior
Posteriora
literature were not measured along the same direction, raising the
possibility 0.09470.021
that
Superiora
0.14670.029
cartilage could be linear elastic (having
the
same
moduli
in
tension
and
compresAverage
0.1167 0.043
sion across the strain origin) but highly
anisotropic (accounting
fore the larger moduli
Middle
Anterior
0.11770.056
Center
0.19070.047
parallel to the surface and smaller moduli perpendicular to
the esurface). This issue
e
Inferior
0.13270.036
was first resolved by Jurvelin et al. in 2003 [52], who reported
thata the compressive
Posterior
0.16770.069
modulus of human knee cartilage was statistically different, but ofa comparable magSuperior
0.14170.045
nitude, when measured on disks harvested
with their axis perpendicular (∼ 1.2 MPa)
Average
0.14170.048
or parallel (∼ 0.8 MPa) to the articular surface. Similar findings were subsequently
a
0.14470.027
reported byGlenoid
Wang et al. in 2003 [86] Superficial
and Chahine et al. in Inferior
2004 [32],
who measured
a
Superior
0.13670.088
the compressive and tensile properties of bovine articular cartilage cubes tested
Average
0.13870.062
along three orthogonal directions. Evidence of the large disparity
between tensile
Middle
Inferiore
0.19570.110
and compressive properties of articular cartilage is shown
in
the
a stress-strain reSuperior
0.16870.094
sponses reported by Huang et al. [50]Average
for human shoulder cartilage (Fig. 19). Today,
0.17870.094
based on the preponderance of evidence reviewed here, a fiber-reinforced elastic
Key: HA0 ; HA0:16 : MPa; n: a ¼ 5; e ¼ 4:
solid matrix is considered
the preferred modeling approach for articular cartilage.
3.5
3
Stress (MPa)
2.5
2
C
D
1
0.5
-0.5
-0.5
0.17670.029
0.18070.105
0.17870.073
0.23370.126
0.18570.121
0.20370.116
k0 ( " 10#14)
B
1.5
0
0.13770.040
0.10170.039
0.16670.075
0.11670.027
0.17570.036
0.14170.051
0.16970.053
0.22670.053
0.15870.042
0.20070.083
0.16970.053
0.17570.052
Table 3
Permeability coefficients of human glenohumeral c
regions and zones
A
A: Tensile (parallel, surface zone)
B: Tensile (perpendicular, surface zone)
C: Tensile (parallel, middle zone)
D: Tensile (perpendicular, middle zone)
E: Compressive (surface zone)
F: Compressive (middle zone)
HA0:16
E
Mating surfaces
Glenoida
Humeral headb
1.3570.82
1.1470.77
Humeral head (region)
Superiorf
Inferiord
Anteriord
Posteriorf
Centerd
0.8770.44
1.0070.62
0.9870.39
1.0370.57
1.8271.27
Glenoid (region)
Superiorf
Inferiore
1.4671.00
1.2270.56
F
-0.4
-0.3
-0.2
-0.1
Strain
0.0
0.1
0.2
Fig. 8. Typical plots of equilibrium stress vs. strain from tensile and
Fig. 19: Equilibrium
stress-strain responses for representative human shoulder
car- head (zone)
Humeral
confined compression tests for the central region of the humeral head.
Z
Superficial
tilage samples
tested
in
tension
and
compression,
as
reported
by
Huang
et
al.
[50]
It provides a good summary of the anisotropy, inhomogeneity, and
i
Middle
(reproducedtension–compression
with permission).nonlinearity
Specimensof tested
tension were
harvested in the
human in
glenohumeral
cartilage,
with the
of the stress–strain
response or
representing
the modulus.
plane tangential
to slope
the articular
surface, parallel
perpendicular
to the localGlenoid
split- (zone)
Note that the tensile portion of the stress–strain curve represents an
line direction.
Superficialf
unconfined tissue response, whereas the compressive portion represents a confined tissue response. The direction of testing (parallel to the
surface for tensile testing, perpendicular to the surface for compression
testing) is also not the same.
Middlee
1.1070.94
1.1770.54
1.1870.51
1.5771.10
Key: k0 : m4/Ns; n: a ¼ 18; b ¼ 47; e ¼ 8; d ¼ 9; Z ¼
6. Discussion
from the superficial zone and 15% of specimens from
the middle zone. It should be noted that failure occurred
in specimens from the superficial zone of the glenoid
(50%) more than those from the superficial zone of the
humeral head (32%).
The complexity of the mechanica
human GHJ articular cartilage was inv
study that addressed the anisotropy,
and nonlinearity of the equilibrium elast
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
43
4 Other Related Mixture Models
This chapter focused on a review of a biphasic mixture of intrinsically incompressible solid and fluid constituents, with a review of selected studies that validated
the model against experimental measurements, mostly in articular cartilage. Since
biphasic theory provides a framework for modeling the solid matrix stresses and
interstitial fluid flow within a porous deformable material, many of the validation
studies employed measurements of the interstitial fluid pressure or flux in response
to mechanical loading to provide direct evidence in support of theoretical predictions. Fundamentally, mixture theory recovers the classical equations of elasticity
theory in the limit when the fluid pressure is set to zero. In the limit when the porous
solid matrix is rigid, it also recovers Darcy’s law, a well-attested phenomenological
relation for flow through porous media. Therefore, the validation studies reported
here effectively demonstrate that the theory can also predict the coupling between
solid matrix deformation and interstitial fluid pressurization.
4.1 Modeling Solutes in Mixtures
Mixture theory has also been extended to include solute transport within the interstitial fluid of a solid-fluid mixture. Extending the theory to incorporate solutes is
relatively straightforward, based on the governing equations covered in Section 2.
When solutes are included, the list of state variables must be extended to include
the solute apparent density ρrα = Jρ α (mass of solute α per volume of the mixture
in the reference configuration). The dependence of the mixture free energy Ψr on
solute density is then embodied in the chemical potential µ α = ∂Ψr /∂ ρrα . If solutes
and the solid matrix are electrically charged, an electric potential ψ may arise if it is
assumed that the mixture must satisfy the electroneutrality condition (much like the
pressure p arises from the assumption that the mixture constituents are intrinsically
incompressible). The fluid pressure p, electrical potential ψ, and chemical potential
µ α may then be combined into a single scalar variable µ̃ α called the mechanoelectrochemical potential. Then, according to the momentum equations, solvent and
solute fluxes are driven by gradients in µ̃ α , as well as inertia and body forces, and
resisted by dissipative momentum exchanges p̂αd similar to the presentation in Section 3.1 and Eq.(20).
The first extension of biphasic theory to include solutes was presented by Lai
et al. in 1991 [57], who modeled cartilage as a triphasic mixture consisting of a
charged porous deformable solid matrix, and an interstitial fluid consisting of a neutral solvent (water) and two monovalent counter-ions from a dissolved salt (such as
Na+ and Cl− ). These authors showed that mixture theory could reproduce Fick’s
law of diffusion in the limit of a free fluid solution, which arises from the momentum balance for the solute. It could also reproduce Donnan’s law to predict the osmotic swelling pressure arising from soluble charge segregation between the triphasic mixture and its surrounding fluid environment. More generally, in addition to
44
Gerard A. Ateshian
permeation, diffusion and barophoresis, electrokinetic phenomena could also be recovered from triphasic theory, such as electrophoresis, electro-osmosis, and streaming potentials and currents [43, 58]. The triphasic formulation was later extended
by Gu et al. in 1998 [44] to include any number of electrolytes. These formulations
demonstrated that mixture theory provides the foundation for modeling a wide range
of phenomena encountered in biological tissues and cells.
Mixture theory provides a fundamental framework that accounts for interactions
among all mixture constituents. Consequently, when classical phenomenological
relations emerge from the mixture equations, such as Darcy’s law and Fick’s law, we
may discover (or rediscover) terms that were neglected in these earlier formulations.
In 2003, Mauck et al. [67] formulated a mixture framework for a neutral solute
in a porous deformable hydrated solid matrix. While mixture theory accounts for
frictional drag tensors fαβ between every pair of constituents, the earlier triphasic
[57] and multi-electrolyte [44] models opted to neglect the friction between solutes
and the solid matrix, arguably because the resulting expressions were sufficient to
reproduce Darcy’s law and Fick’s law. By keeping the frictional drag between solute
and solid, Mauck et al. [67] showed that the resulting momentum equations for the
solute could differentiate between solute diffusivity within a free fluid versus the
diffusivity within the mixture (inclusive of the solid matrix). While experiments had
long attested that these diffusivities could be different (depending on the molecular
size of the solute relative to the pore structure of the solid) [40], most models of
solute transport within porous media simply employed Fick’s phenomenological
law with an adjusted value of the solute diffusivity.
Mauck et al. [67] showed that the mixture formulation could predict phenomena
resulting from the interaction of solid matrix deformation and solute transport that
were not anticipated by the phenomenological relations. Most notably, they found
that dynamic loading of a disk of tissue, or hydrogel, submerged in a bath containing a solute, would increase the solute concentration far above that predicted from
Fick’s law. A series of subsequent experimental studies validated these predictions
in agarose and articular cartilage [3, 5, 6, 33], providing further confidence that
mixture theory is a sound framework for extending classical formulations (Fig. 20).
Similarly, using basic principles from mixture theory and physical chemistry, Mauck
et al. [67] proposed a formulation for solute partitioning between the pore space in
the mixture and a surrounding solution, which could account for solid matrix deformation and incomplete volume recovery in response to osmotic loading. This
formulation was later found to accurately predict the response of hydrogels and
chondrocytes to osmotic loading using a variety of osmolytes [2, 4, 15].
4.2 Constrained Solid Mixtures
As shown by Humphrey and Rajagopal [51], mixture theory may also be used to
model the solid matrix of biological tissues that have heterogeneous constituents,
such as mixtures of collagen, elastin, and smooth muscle cells, as found in the aor-
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
45
20
9.2%
6.9%
5.9%
16
ĉ
12
8
4
0
0
10
20
30
time (h)
40
Fig. 20: Experimental results (symbols) and theoretical predictions (solid curves)
for the uptake of 70 kDa dextran into agarose disks of various concentrations (see
legend) in response to dynamic unconfined compression (±5 % strain amplitude
at 1 Hz, superimposed on a 15 % compressive strain offset [3]) for 40 h. ĉ is the
ratio of average dextran concentration in the dynamically loaded disk to the average
concentration achieved at steady state in the absence of loading; it represents the
enhancement ratio resulting from dynamic loading. Theoretical predictions were
obtained by independently measuring the mechanical and transport properties of
this agarose-dextran system and using them in the mixture model formulated by
Mauck et al. [67]. Reproduced from the study by Albro et al. [5] (with permission).
tic wall. When these constituents are physically bound together, it may be assumed
that the solid mixture is constrained such that all solid constituents α share the same
velocity vα ≡ vs . It is noteworthy that this assumption simplifies the formulation of
the mixture governing equations, since the diffusion velocity uα and dissipative
momentum supply p̂αd (Section 2.2) reduce to zero in the case of such mixtures.
Moreover, it may be assumed that each solid constituent in the mixture has a different reference configuration Xα [13, 85], even though all constituents share the same
current configuration x, which makes it possible to model the evolution of residual
stresses in growing tissues.
In hindsight, another model commonly adopted in the biomechanics of biological
soft tissues implicitly describes constrained solid mixtures whose constituents all
share the same reference configuration. Notably, Lanir’s approach for modeling soft
tissues using continuous fiber distributions [59, 60] relies on the superposition of
fiber bundles that may contribute to the tissue response only if their normal strain is
tensile, as illustrated in Eq.(82). In his approach, which has been widely followed
in the biomechanics literature, only fiber bundles that are in tension contribute to
the response; thus, each fiber bundle may be viewed as a mixture constituent, even
while the number of constituents in the mixture varies with the state of loading. Yet,
all fibril bundles share the same reference and current configuration, satisfying the
assumption of a constrained solid mixture.
46
Gerard A. Ateshian
4.3 Growth and Remodeling
As shown in the mass balance equation (1) for each constituent, mixture theory allows mass exchanges, thus chemical reactions, between its constituents. Reactions
that add or remove mass from the solid matrix of a biological tissue may describe
growth and remodeling. An elegant demonstration of this basic concept was presented by Cowin and Hegedus in 1976 [36], who modeled interstitial growth and
remodeling of bone in response to mechanical loading using a mass supply term
for the solid matrix, which depended on the state of strain. The supply of mass to
the solid matrix came from implicit soluble constituents available in the trabecular
pore space. Modeling growth using mixture theory has since been reprised by many
authors, as reviewed in [7, 12], and is currently an active topic of investigation.
It is intriguing that the theory of reactive mixtures may also be used to model
classical phenomena such as viscoelasticity, where the viscous behavior results from
bonds that break in response to loading, and reform in a stress-free state [11]. Other
related phenomena, such as damage mechanics, may thus be similarly modeled,
where bonds break permanently in response to loading. A main advantage of this
approach is that bond mass densities are observable variables whose temporal evolution is governed by the equation of mass balance. Since the evolving composition
of a material may be measured experimentally, reactive mixture models may be validated directly against such measurements.
5 Summary
Mixture theory is an elegant continuum mechanics framework that is well suited for
modeling biological tissues. While many biomechanics investigators have opted to
use this framework since its introduction in the 1960s, it is fair to state that some
of its earliest and most determined proponents were Van C. Mow and W. Michael
Lai,1 who applied it to the modeling of articular cartilage, and worked over several decades in an effort to validate and extend this framework to accommodate the
complexities of biological tissues. With their biphasic theory, they applied mixture
theory to model biological tissues as deformable porous media.
Porous media had already been successfully modeled using earlier theories, first
proposed by Fillunger, then Terzaghi, as reviewed in the captivating historical perspective of de Boer [39], subsequently advanced by Biot [25] as consolidation theory, and recast as poroelasticity theory by Rice and Cleary [76]. Indeed, Mow et
al. [68] and Bowen [28] pointed out the equivalence of their approaches to Biot’s
earlier work. However, despite its elegance, the poroelasticity framework did not
provide the foundations for extending the theory to multiple constituents, nor mass
exchanges among the constituents. By using mixture theory, Lai and Mow showed
that biphasic theory could be systematically extended to triphasic theory [57], which
1
Mow and Lai were this author’s doctoral advisors and mentors, starting in 1986.
Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage
47
laid the foundations for many subsequent developments, as partially reviewed in
Section 4.
Many other investigators have independently demonstrated the value of mixture
theory for modeling complex phenomena in biological tissues, as also partially reviewed above. Because theoretical frameworks are critically dependent on constitutive assumptions, there is no unique formulation of mixture theory for a given combination of fluid and solid constituents. Consequently, I felt that an exhaustive review and comparison of mixture models in biological tissues was beyond the scope
of this chapter. The modeling approach reviewed here provides the foundation of the
framework I have found useful in my own investigations and applications to biological tissues. In collaboration with Jeffrey Weiss, Steve Maas and other colleagues,
we have strived to provide finite element computational tools [63] that implement
biphasic and multiphasic theories [17, 19], including contact mechanics [16, 18] and
chemical reactions [20], to facilitate the dissemination of this framework within the
biomechanics community.
Acknowledgments
I would like to thank the many former and current doctoral students and fellows
who labored with me over the years, performing experiments and validating the
mixture models used in our investigations of biological tissues: Dr. Huiqun (Laura)
Wang, Dr. William H. Warden, Dr. Michael A. Soltz, Prof. Robert L. Mauck, Prof.
Chun-Yuh (Charles) Huang, Dr. Changbin Wang, Dr. Ramaswamy Krishnan, Prof.
Seonghun Park, Prof. Ines M. Basalo, Prof. Nadeen O. Chahine, Dr. Michael B. Albro, Dr. Matteo M. Caligaris, Dr. Clare Canal-Guterl, Dr. Sevan R. Oungoulian, Mr.
Alexander D. Cigan, Mr. Robert J. Nims, Mr. Brian K. Jones, Mr. Chieh Hou, and
Ms. Krista M. Durney. I would also like to thank Dr. Albro for providing the experimental data appearing in Fig. 4, and Mr. Brandon K. Zimmerman for re-analyzing
older data sets to produce Fig. 13.
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